Congruence properties of the binary partition function

1975 ◽  
Vol 78 (3) ◽  
pp. 437-442 ◽  
Author(s):  
M. D. Hirschhorn ◽  
J. H. Loxton
Keyword(s):  
Partition Function ◽  
Binary Partition ◽  
Powers Of 2 ◽  
Congruence Properties

We denote by b(n) the number of binary partitions of n, that is the number of partitions of n as the sum of powers of 2. As usual, two partitions are not considered to be distinct if they differ only in the order of their summands. It is convenient to define b(0) = 1.

Congruence properties of the binary partition function

1969 ◽  
Vol 66 (2) ◽  
pp. 371-376 ◽  
Author(s):  
R. F. Churchhouse
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Functional Equations ◽  
Precise Calculation ◽  
Binary Partition ◽  
Powers Of 2 ◽  
De Bruijn ◽  
Congruence Properties

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

scholarly journals Congruence Properties of Binary Partition Functions

2013 ◽  
Vol 17 (1) ◽  
pp. 15-26 ◽  
Author(s):  
Katherine Anders ◽  
Melissa Dennison ◽  
Jennifer Weber Lansing ◽  
Bruce Reznick
Keyword(s):  
Partition Functions ◽  
Binary Partition ◽  
Congruence Properties

scholarly journals On congruence properties of the partition function

2000 ◽  
Vol 23 (7) ◽  
pp. 493-496 ◽  
Author(s):  
Jayce Getz
Keyword(s):  
Partition Function ◽  
Congruence Properties

Some congruence properties of the partition function are proved.

Ramanujan Congruences for p-k(n)

1968 ◽  
Vol 20 ◽  
pp. 67-78 ◽  
Author(s):  
A. O. L. Atkin
Keyword(s):  
Partition Function ◽  
Ramanujan Congruences ◽  
Image Position ◽  
Congruence Properties

Let12Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

Congruences modulo powers of 2 for a certain partition function

2010 ◽  
Vol 22 (1) ◽  
pp. 101-117 ◽  
Author(s):  
Hei-Chi Chan ◽  
Shaun Cooper
Keyword(s):  
Partition Function ◽  
Powers Of 2
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