A175239 Number of AP divisors of n.

1, 2, 3, 3, 3, 4, 3, 4, 4, 5, 3, 6, 3, 5, 5, 5, 3, 7, 3, 6, 6, 5, 3, 8, 4, 5, 6, 6, 3, 9, 3, 6, 6, 5, 5, 10, 3, 5, 6, 8, 3, 9, 3, 7, 8, 5, 3, 10, 4, 7, 6, 7, 3, 9, 6, 8, 6, 5, 3, 13, 3, 5, 8, 7, 6, 9, 3, 7, 6, 9, 3, 12, 3, 5, 9, 7, 5, 10, 3, 10, 7, 5, 3, 13, 6, 5, 6, 8, 3, 14, 5, 7, 6, 5, 6, 12, 3, 7, 8, 10

Offset

1,2

Comments

The following definition is given in the reference: k is an AP divisor of n if there exists a partition of n that is an arithmetic progression of length k; arithmetic progressions of length 1 or greater are counted.

Terms 1-30 were given in the reference; others were calculated from the generating function by the author.

Links

Table of n, a(n) for n=1..100.

Augustine O. Munagi, Combinatorics of Integer Partitions in Arithmetic Progression, Integers, A7, Volume 10 (2010), 73-82.

Formula

G.f.: sum(k>=1, q^k*(1+q^k+q^(2*k^2))/(1-q^(2*k)) ).

Example

The partitions of 4 that are arithmetic progressions are (4), (2,2), (3,1) and (1,1,1,1) with lengths 1, 2, 2 and 4, respectively. The AP divisors of 4 are thus 1, 2 and 4, so a(4)=3. - Corrected by Jaroslav Krizek, Mar 26 2010

Crossrefs

Sequence in context: A322988 A098201 A341883 * A176228 A322822 A305232

Adjacent sequences: A175236 A175237 A175238 * A175240 A175241 A175242

Keyword

nonn

Author

John W. Layman, Mar 10 2010

Status

approved