Exceptional congruences for powers of the partition function

Boylan, Matthew

doi:10.4064/aa111-2-7

Cited by 35 publications

(27 citation statements)

References 2 publications

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“…The upper bound on implied by Theorem 1.2 is very close to being sharp in j and is sharp in N : this follows from the unexceptional congruences for even powers of the generating function for p(n) reported on in [10], the exceptional Ramanujan congruences (mod ) for coefficients of odd powers of the generating function of p(n) as reported on in [4] and a line of elementary algebra to use the latter Ramanujan congruences to produce Ramanujan congruences for c(n) with j even.…”

Section: Introductionmentioning

confidence: 66%

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Ramanujan Congruences for a Class of Eta Quotients

Sinick

2010

Int. J. Number Theory

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We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes for which their coefficients c(n) obey congruences of the form c( n + a) ≡ 0 (mod ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences c N (n) satisfy, where c N (n) is defined by(1−q n )(1−q N n ) for N > 1. This last result answers a question of H.-C. Chan.

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Section: Introductionmentioning

confidence: 66%

“…It is natural to ask whether p(n) satisfies any other congruences of the same form. In [2], Ahlgren and Boylan showed that Ramanujan's congruences are the only congruences of this form: if is prime, 0 ≤ a ≤ − 1 and p( n + a) ≡ 0 (mod ), then ( , a) ∈ { (5,4), (7,5), (11,6)}.…”

Section: Introductionmentioning

confidence: 99%

Ramanujan Congruences for a Class of Eta Quotients

Sinick

2010

Int. J. Number Theory

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“…This can be seen in papers by Andrews [6], Atkin [7], Kiming and Olsson [10], Serre [13], Newman [11], Boylan [8], and Gandhi [9], to name a few.…”

Section: Congruences For Multipartitionsmentioning

confidence: 89%

A Combinatorial Proof of a Recursive Formula for Multipartitions

Lazarev¹,

Swisher²

2012

Integers

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For k 1, let p k .n/ count the number of k-component multipartitions of a nonnegative integer n, and let .n/ D P d jn d be the usual divisor function. In this paper, we give a combinatorial proof of the recursive formula p k .n/ D k n n X rD1 p k .n r/ .r/;both for k 1, where p k .n/ is defined as above, and also for k < 0, which requires a subtler approach. This formula was used by Gandhi in 1963 to prove several theorems, which yield numerous Ramanujan type congruences for p k .n/, including some well-known congruences for Ramanujan's -function.

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“…All the pairs (r, ) with exceptional for r < 48 are listed in [6]. We will look at the pairs in the following set S = (r, ) exceptional for r, r 24 ∪ (r, ) exceptional for r, 24 < r < 48 and 24 − r = −r .…”

Section: Conjecture 13 (Seementioning

confidence: 99%

Distribution of powers of the partition function modulo ℓj

Brown

2009

Journal of Number Theory

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In this paper we study Newman's conjecture for powers of the partition function. While this conjecture is known for powers of primes that are not exceptional for the power under consideration, it is an open problem for exceptional primes. We settle this conjecture in many cases for small powers of the partition function by generalizing results of Ono and Ahlgren. It should be noted our method requires a case by case examination of each power and does not yield a general method for dealing with different powers simultaneously.Published by Elsevier Inc.

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Exceptional congruences for powers of the partition function

Cited by 35 publications

References 2 publications

Ramanujan Congruences for a Class of Eta Quotients

Ramanujan Congruences for a Class of Eta Quotients

A Combinatorial Proof of a Recursive Formula for Multipartitions

Distribution of powers of the partition function modulo ℓj

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