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1

ROSEN, JULIAN. "MULTIPLE HARMONIC SUMS AND WOLSTENHOLME'S THEOREM." International Journal of Number Theory 09, no. 08 (2013): 2033–52. http://dx.doi.org/10.1142/s1793042113500735.

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We give a family of congruences for the binomial coefficient [Formula: see text], with k an integer and p a prime. Our congruences involve multiple harmonic sums, and hold modulo arbitrary large powers of p. The general congruence in our family, which depends on a parameter n, involves n "elementary symmetric" multiple harmonic sums, and holds modulo p2n+3. These congruences are actually part of a much larger collection of congruences for [Formula: see text] in terms of the elementary symmetric multiple harmonic sums. Congruences in our family have been optimized, in that they involve the fewe
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2

Shen, Erin Y. Y. "Congruences modulo 9 for singular overpartitions." International Journal of Number Theory 13, no. 03 (2017): 717–24. http://dx.doi.org/10.1142/s1793042117500361.

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In a recent work, Andrews introduced the new combinatorial objects called singular overpartitions. He proved that these singular overpartitions can be enumerated by the partition function [Formula: see text] which denotes the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. In this paper, we consider the function [Formula: see text] from an arithmetical point of view. We establish a number of Ramanujan-like congruences and a congruence relation modulo [Formula: see text] for [Formula: see t
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3

RADU, CRISTIAN-SILVIU, and JAMES A. SELLERS. "CONGRUENCES MODULO SQUARES OF PRIMES FOR FU'S k DOTS BRACELET PARTITIONS." International Journal of Number Theory 09, no. 04 (2013): 939–43. http://dx.doi.org/10.1142/s1793042113500073.

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In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by 𝔅k(n) and proved
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4

Garvan, F. G. "More cranks and t-cores." Bulletin of the Australian Mathematical Society 63, no. 3 (2001): 379–91. http://dx.doi.org/10.1017/s0004972700019481.

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Dedicated to George Szekeres on the occasion of his 90th BirthdayIn 1990, new statistics on partitions (called cransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.
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5

Yao, Olivia X. M. "Infinite families of congruences modulo 5 and 7 for the cubic partition function." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 5 (2019): 1189–205. http://dx.doi.org/10.1017/prm.2018.61.

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AbstractIn 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8}
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6

Meštrović, Romeo. "An Extension of a Congruence by Tauraso." ISRN Combinatorics 2013 (December 25, 2013): 1–7. http://dx.doi.org/10.1155/2013/363724.

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For a positive integer let be the th harmonic number. In this paper we prove that, for any prime , . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers.
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7

CHAN, HENG HUAT, LIUQUAN WANG, and YIFAN YANG. "CONGRUENCES MODULO 5 AND 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS." Journal of the Australian Mathematical Society 103, no. 2 (2016): 157–76. http://dx.doi.org/10.1017/s1446788716000616.

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Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.
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8

Gouri Shankar Guru. "On Congruences Modulo Powers of 2 for (2, 4)-Regular Overpartitions." Advances in Nonlinear Variational Inequalities 28, no. 7s (2025): 24–36. https://doi.org/10.52783/anvi.v28.4481.

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Recently, Naika et al. (2021) proved several congruence properties modulo powers of 2 for (j,k)-regular overpartition of n for, and . This study establishes several infinite families of congruences modulo powers of 2 for employing q-series identities and iterative computational techniques.
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9

Yao, Olivia X. M. "Arithmetic properties for Fu's 9 dots bracelet partitions." International Journal of Number Theory 11, no. 04 (2015): 1063–72. http://dx.doi.org/10.1142/s1793042115500566.

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The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let 𝔅k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 𝔅5(n), 𝔅7(n) and 𝔅11(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for 𝔅5(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for 𝔅9(n
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10

Jin, Hai-Tao, та Li Zhang. "Ramanujan-type congruences for ℓ-regular partitions modulo 3, 5, 11 and 13". International Journal of Number Theory 13, № 08 (2017): 1995–2006. http://dx.doi.org/10.1142/s179304211750107x.

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Let [Formula: see text] be the number of [Formula: see text]-regular partitions of [Formula: see text]. Recently, Hou et al. established several infinite families of congruences for [Formula: see text] modulo [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for [Formula: see text] modulo [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text], we obtain three infinite families of congruences for [Formula: see text] modulo [Formula: see text] a
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11

CUI, SU-PING, and NANCY SHAN SHAN GU. "CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS." International Journal of Number Theory 09, no. 08 (2013): 1885–94. http://dx.doi.org/10.1142/s1793042113500644.

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Andrews and Paule introduced broken k-diamond partitions by using MacMahon's partition analysis. Recently, Fu found a generalization which he called k dots bracelet partitions and investigated some congruences for this kind of partitions. In this paper, by finding congruence relations between the generating function for 5 dots bracelet partitions and that for 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime p, we study arithmetic properties modulo p of k dots bracelet partitions.
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12

Kohnen, Winfried. "Some Congruences Modulo Primes." Monatshefte f�r Mathematik 127, no. 4 (1999): 321–24. http://dx.doi.org/10.1007/s006050050043.

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13

JAMESON, MARIE. "A REFINEMENT OF RAMANUJAN'S CONGRUENCES MODULO POWERS OF 7 AND 11." International Journal of Number Theory 08, no. 04 (2012): 865–79. http://dx.doi.org/10.1142/s1793042112500510.

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Ramanujan's famous congruences for the partition function modulo powers of 5, 7, and 11 have inspired much further research. For example, in 2002 Lovejoy and Ono found subprogressions of 5jn + β5(j) for which Ramanujan's congruence mod 5j could be strengthened to a statement modulo 5j+1. Here we provide the analogous results modulo powers of 7 and 11. We require the arithmetic properties of two special elliptic curves.
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14

Shen, Zhongyan. "Congruences Involving Special Sums of Triple Reciprocals." Journal of Mathematics 2024 (February 2, 2024): 1–8. http://dx.doi.org/10.1155/2024/8445635.

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Define the sums of triple reciprocals Zn=∑i+j+k=n1/ijk,i,j,k≥1. Zhao discovered the following curious congruence for any odd prime p, Zp≡−2Bp−3mod p. Xia and Cai extended the above congruence to modulo p2,Zp≡12Bp−3/p−3−3B2p−4/p−2mod p2, where p>5 is a prime. In this paper, we consider the congruences about Zp−1+N/N (where x is the integral part of x, N=1,2,3,4,6) modulo p2. When N=1, the results we obtain are the results of Zhao and Xia and Cai.
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15

Guru, Gouri Shankar, and Yudhisthira Jamudulia. "SOME INFINITE FAMILIES OF CONGRUENCESFOR OVERPARTITIONS WITH RESTRICTEDODD DIFFERENCES." JP Journal of Algebra, Number Theory and Applications 64, no. 5 (2025): 531–45. https://doi.org/10.17654/0972555525027.

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Recently, Hanson and Smith [7] proved Ramanujan type congruences modulo 3 for , where represents the number of overpartitions of with restricted odd differences. They also derived congruences modulo 5 for . In this paper, we prove several infinite families of congruences modulo powers of 2 for by employing -series identities and iterative computations.
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16

Ahmed, Zakir, and Nayandeep Deka Baruah. "New congruences for Andrews' singular overpartitions." International Journal of Number Theory 11, no. 07 (2015): 2247–64. http://dx.doi.org/10.1142/s1793042115501018.

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Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function [Formula: see text] which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i ( mod k) may be overlined. He also proved that [Formula: see text]. Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we find
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17

LIN, BERNARD L. S., and ANDREW Y. Z. WANG. "GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES." Bulletin of the Australian Mathematical Society 90, no. 2 (2014): 204–12. http://dx.doi.org/10.1017/s0004972714000343.

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AbstractRecently, Keith used the theory of modular forms to study 9-regular partitions modulo 2 and 3. He obtained one infinite family of congruences modulo 3, and meanwhile proposed an analogous conjecture. In this note, we show that 9-regular partitions and 3-cores satisfy the same congruences modulo 3. Thus, we first derive several results on 3-cores, and then generalise Keith’s conjecture and get a stronger result, which implies that all of Keith’s results on congruences modulo 3 are consequences of our result.
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18

Rosen, Julian. "A general family of congruences for Bernoulli numbers." International Journal of Number Theory 14, no. 07 (2018): 1895–902. http://dx.doi.org/10.1142/s1793042118501129.

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We prove a general family of congruences for Bernoulli numbers whose index is a polynomial function of a prime, modulo a power of that prime. Our family generalizes many known results, including the von Staudt–Clausen theorem and Kummer’s congruence.
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19

Straub, Armin. "Congruences for Fishburn numbers modulo prime powers." International Journal of Number Theory 11, no. 05 (2015): 1679–90. http://dx.doi.org/10.1142/s1793042115400175.

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The Fishburn numbers ξ(n) are defined by the formal power series [Formula: see text] Recently, Andrews and Sellers discovered congruences of the form ξ(pm + j) ≡ 0 modulo p, valid for all m ≥ 0. These congruences have then been complemented and generalized to the case of r-Fishburn numbers by Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend to hold modulo prime powers.
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20

Sun, Zhi-Wei, and Donald M. Davis. "Combinatorial congruences modulo prime powers." Transactions of the American Mathematical Society 359, no. 11 (2007): 5525–53. http://dx.doi.org/10.1090/s0002-9947-07-04236-5.

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21

Zhao, Li-Lu, and Zhi-Wei Sun. "Some curious congruences modulo primes." Journal of Number Theory 130, no. 4 (2010): 930–35. http://dx.doi.org/10.1016/j.jnt.2009.11.010.

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22

Yang, Yining, and Peng Yang. "Congruences for harmonic sums." Notes on Number Theory and Discrete Mathematics 29, no. 1 (2023): 137–46. http://dx.doi.org/10.7546/nntdm.2023.29.1.137-146.

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Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p^2. In this paper, we improve the harmonic sums \[ H_{p}(n)=\sum\limits_{\substack{l_{1}+l_{2}+\cdots+l_{n}=p\\ l_{1}, l_{2}, \ldots , l_{n}>0}} \frac{1}{l_{1} l_{2} \cdots l_{n}} \] to supercongruences modulo p^3 and p^4 for odd and even where prime p>8 and 3 \leq n \leq p-6.
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23

Al-Shaghay, Abdullah, and Karl Dilcher. "Analogues of the binomial coefficient theorems of Gauss and Jacobi." International Journal of Number Theory 12, no. 08 (2016): 2125–45. http://dx.doi.org/10.1142/s179304211650127x.

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The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a s
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24

Das, Sabuj. "“RAMANUJAN’S CONGRUENCES AND DYSON’S CRANK”." International Journal of Research -GRANTHAALAYAH 2, no. 3 (2014): 10–32. http://dx.doi.org/10.29121/granthaalayah.v2.i3.2014.3056.

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In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11. Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famo
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25

Sabuj, Das. ""RAMANUJAN'S CONGRUENCES AND DYSON'S CRANK"." International Journal of Research - GRANTHAALAYAH 2, no. 3 (2017): 10–32. https://doi.org/10.5281/zenodo.884058.

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In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11. Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famo
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26

MEŠTROVIĆ, ROMEO. "AN EXTENSION OF SURY’S IDENTITY AND RELATED CONGRUENCES." Bulletin of the Australian Mathematical Society 85, no. 3 (2011): 482–96. http://dx.doi.org/10.1017/s0004972711002826.

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AbstractIn this paper we give an extension of a curious combinatorial identity due to B. Sury. Our proof is very simple and elementary. As an application, we obtain two congruences for Fermat quotients modulo p3. Moreover, we prove an extension of a result by H. Pan that generalizes Carlitz’s congruence.
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27

XIA, ERNEST X. W., and OLIVIA X. M. YAO. "CONGRUENCES MODULO POWERS OF 2 FOR FU’S 5 DOTS BRACELET PARTITIONS." Bulletin of the Australian Mathematical Society 89, no. 3 (2013): 360–72. http://dx.doi.org/10.1017/s0004972713000737.

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AbstractIn 2007, Andrews and Paule introduced a new class of combinatorial objects called broken $k$-diamond partitions. Recently, Shishuo Fu generalised the notion of broken $k$-diamond partitions to combinatorial objects which he termed $k$ dots bracelet partitions. Fu denoted the number of $k$ dots bracelet partitions of $n$ by ${\mathfrak{B}}_{k} (n)$ and proved several congruences modulo primes and modulo powers of 2. More recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for ${\mathfrak{B}}_{5} (n)$, ${\mathfrak{
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28

Elkhiri, Laid, Miloud Mihoubi, and Abdellah Derbal. "Congruences involving alternating sums related to harmonic numbers and binomial coefficients." Notes on Number Theory and Discrete Mathematics 26, no. 4 (2020): 39–51. http://dx.doi.org/10.7546/nntdm.2020.26.4.39-51.

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In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number p (super congruences) in the ring of p-integer \mathbb{Z}_{p} involving binomial coefficients and generalized harmonic numbers.
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29

McCoy, Megan, Kevin Thielen, Liuquan Wang, and Jianqiang Zhao. "A family of super congruences involving multiple harmonic sums." International Journal of Number Theory 13, no. 01 (2016): 109–28. http://dx.doi.org/10.1142/s1793042117500075.

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In recent years, the congruence [Formula: see text] first discovered by the last author has been generalized by either increasing the number of indices and considering the corresponding super congruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar super congruences modulo prime powers [Formula: see text] with the indices summing up to [Formula: see text] where [Formula: see text] is coprime to [Formula: see text], and where all the indices are also coprime to [Formula: see text].
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30

Liu, Eric H., and Wenjing Du. "Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions." Open Mathematics 17, no. 1 (2019): 356–64. http://dx.doi.org/10.1515/math-2019-0026.

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Abstract Singular overpartition functions were defined by Andrews. Let Ck,i(n) denote the number of (k, i)-singular overpartitions of n, which counts the number of overpartitions of n in which no part is divisible by k and only parts ±i (mod k) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for Ck,i(n) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs (k, i). In this paper, we prove some congruences modulo powers of 2 for C48, 6(n) and C48, 18(n).
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31

Wagstaff, Samuel S. "Congruences for rs(n) modulo 2s." Journal of Number Theory 127, no. 2 (2007): 326–29. http://dx.doi.org/10.1016/j.jnt.2007.05.002.

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32

HESSAMI PILEHROOD, KH, T. HESSAMI PILEHROOD, and R. TAURASO. "CONGRUENCES CONCERNING JACOBI POLYNOMIALS AND APÉRY-LIKE FORMULAE." International Journal of Number Theory 08, no. 07 (2012): 1789–811. http://dx.doi.org/10.1142/s1793042112501035.

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Let p > 5 be a prime. We prove congruences modulo p3-d for sums of the general form [Formula: see text] and [Formula: see text] with d = 0, 1. We also consider the special case t = (-1)d/16 of the former sum, where the congruences hold modulo p5-d.
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33

Sugandha, Agus, and Irfan Azkamahendra. "General and Recursive Forms of Catalan Numbers and Modulo Prime Catalan Numbers to the Power of Positive Integers." Perwira Journal of Science & Engineering 2, no. 2 (2022): 15–20. http://dx.doi.org/10.54199/pjse.v2i2.133.

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. A Catalan number is a positive number obtained by calculating the combined structure of a sequence. Catalan numbers have a general form and a recursive form that can be identified through Diagonal-Avoiding Paths and Balanced Parentheses. Catalan numbers have congruence on the modulo of integers. One of them is on the prime number modulo p. For every odd prime p, p does not divisible by and the product of all numbers d by d between 0 and and the Greatest Common Divisor of d and p is 1, will be congruent to -1 modulo . For every integer a with a between 0 and , the Catalan numbers have differe
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34

Ray, Chiranjit, and Rupam Barman. "Infinite families of congruences for k-regular overpartitions." International Journal of Number Theory 14, no. 01 (2017): 19–29. http://dx.doi.org/10.1142/s1793042118500021.

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Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we find infinite families of congruences modulo 4, 8 and 16 for [Formula: see text] and [Formula: see text] for any [Formula: see text]. Along the way, we obtain several Ramanujan type congruences for [Formula: see text] and [Formula: see text]. We also find infinite families of congruences modulo [Formula: see text] for [Formula: see text].
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35

Mahadeva Naika, M. S., and C. Shivashankar. "Congruences for Andrews singular overpartition pairs." International Journal of Number Theory 14, no. 04 (2018): 989–1008. http://dx.doi.org/10.1142/s1793042118500586.

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Andrews defined the combinatorial objects called singular overpartitions denoted by [Formula: see text], which counts the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. In this paper, we investigate the arithmetic properties of Andrews singular overpartition pairs. Let [Formula: see text] be the number of overpartition pairs of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. We will prove a number of Ramanujan l
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36

GARVAN, FRANK G., and CHRIS JENNINGS-SHAFFER. "HECKE-TYPE CONGRUENCES FOR ANDREWS' SPT-FUNCTION MODULO 16 AND 32." International Journal of Number Theory 10, no. 02 (2014): 375–90. http://dx.doi.org/10.1142/s1793042113500991.

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Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt (n). We show that a normalized form of the generating function of spt (n) is an eigenform modulo 32 for the Hecke operators T(ℓ2) for primes ℓ ≥ 5 with ℓ ≡ 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for ℓ ≡ 13, 23 (mod 24).
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37

Guo, Victor, and Michael Schlosser. "Some New q-Congruences for Truncated Basic Hypergeometric Series." Symmetry 11, no. 2 (2019): 268. http://dx.doi.org/10.3390/sym11020268.

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We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews’ multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the f
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38

LIN, BERNARD L. S., JIAN LIU, ANDREW Y. Z. WANG, and JIEJUAN XIAO. "INFINITE FAMILIES OF CONGRUENCES FOR OVERPARTITIONS WITH RESTRICTED ODD DIFFERENCES." Bulletin of the Australian Mathematical Society 102, no. 1 (2020): 59–66. http://dx.doi.org/10.1017/s0004972719001254.

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Let $\overline{t}(n)$ be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for $\overline{t}(n)$ modulo small powers of $2$ and $3$ have been established. We present two infinite families of congruences modulo $5$ and $27$ for $\overline{t}(n)$, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci. 129 (2019), Article 31].
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39

Puneeth, V., та Anirban Roy. "FAMILY OF CONGRUENCES FOR (2, β) REGULAR BIPARTITION TRIPLES". South East Asian J. of Mathematics and Mathematical Sciences 18, № 03 (2022): 01–14. http://dx.doi.org/10.56827/seajmms.2022.1803.1.

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Though congruences have their limitations, they have significant impor- tance in the field of number theory and helps in proving many interesting results. Thus, this article has adopted the technique and properties of congruences to iden- tify and prove a set of congruent properties for integer partition. The partition of a positive integer is a way of expressing the number as a sum of positive in- tegers. One such partitions known as regular bipartition triple are discussed in this article. New congruences modulo even integers and modulo prime (p 5) powers are derived for (2, β) regular bipar
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40

Cui, Su-Ping, Wen Xiang Gu, and Zhen Sheng Ma. "Congruences for partitions with odd parts distinct modulo 5." International Journal of Number Theory 11, no. 07 (2015): 2151–59. http://dx.doi.org/10.1142/s1793042115500943.

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Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct modulo powers of 3. In this paper, we focus on congruences for pod (n) modulo 5. For example, for α ≥ 2 and n ≥ 0, we have [Formula: see text] In addition, applying some equations given by Hirschhorn and Sellers [Arithmetic properties of partitions with odd parts distinct, Ramanujan J.22 (2010) 273–284], some new congruences are established.
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41

YAO, OLIVIA X. M. "INFINITE FAMILIES OF CONGRUENCES MODULO 3 AND 9 FOR BIPARTITIONS WITH 3-CORES." Bulletin of the Australian Mathematical Society 91, no. 1 (2014): 47–52. http://dx.doi.org/10.1017/s0004972714000586.

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AbstractLet $A_{3}(n)$ denote the number of bipartitions of $n$ with 3-cores. Recently, Lin [‘Some results on bipartitions with 3-core’, J. Number Theory139 (2014), 44–52] established some congruences modulo 4, 5, 7 and 8 for $A_{3}(n)$. In this paper, we prove several infinite families of congruences modulo 3 and 9 for $A_{3}(n)$ by employing two identities due to Ramanujan.
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42

Adansie, Prince, Shane Chern, and Ernest X. W. Xia. "New infinite families of congruences for the number of tagged parts over partitions with designated summands." International Journal of Number Theory 14, no. 07 (2018): 1935–42. http://dx.doi.org/10.1142/s1793042118501154.

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Recently, Lin introduced a new partition function [Formula: see text], which counts the total number of tagged parts over all partitions of [Formula: see text] with designated summands. Lin also proved some congruences modulo [Formula: see text] and [Formula: see text] for [Formula: see text]. In this paper, we shall present two new infinite families of congruences modulo [Formula: see text] for [Formula: see text].
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43

Shen, Zhongyan, та Tianxin Cai. "Congruences involving alternating harmonic sums modulo pα qβ". Mathematica Slovaca 68, № 5 (2018): 975–80. http://dx.doi.org/10.1515/ms-2017-0159.

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Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)
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44

Wang, Yunpeng, and Jizhen Yang. "Modulo $p^2$ congruences involving harmonic numbers." Annales Polonici Mathematici 121, no. 3 (2018): 263–78. http://dx.doi.org/10.4064/ap180401-12-9.

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45

Cochrane, Todd. "Small zeros of quadratic congruences modulo pq." Mathematika 37, no. 2 (1990): 261–72. http://dx.doi.org/10.1112/s0025579300012985.

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46

Zhang, Li. "Ramanujan-type congruences for overpartitions modulo $3$." Rocky Mountain Journal of Mathematics 50, no. 6 (2020): 2257–64. http://dx.doi.org/10.1216/rmj.2020.50.2257.

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47

MAHADEVA NAIKA, M. S., B. HEMANTHKUMAR, and H. S. SUMANTH BHARADWAJ. "CONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS." Bulletin of the Australian Mathematical Society 93, no. 3 (2015): 400–409. http://dx.doi.org/10.1017/s0004972715001367.

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Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$. In the process, we also prove numerous parity results for broken 7-diamond partitions.
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48

Sun, Zhi-Hong. "Cubic and quartic congruences modulo a prime." Journal of Number Theory 102, no. 1 (2003): 41–89. http://dx.doi.org/10.1016/s0022-314x(03)00067-2.

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49

Lin, Bernard L. S. "Congruences modulo 27 for cubic partition pairs." Journal of Number Theory 171 (February 2017): 31–42. http://dx.doi.org/10.1016/j.jnt.2016.07.012.

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50

Chen, William Y. C., Lisa H. Sun, Rong-Hua Wang, and Li Zhang. "Ramanujan-type congruences for overpartitions modulo 5." Journal of Number Theory 148 (March 2015): 62–72. http://dx.doi.org/10.1016/j.jnt.2014.09.017.

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