A139807 Number of distinct values of Product_{p is a part of P} (p-1) when P ranges over all partitions of n.

1, 2, 2, 3, 3, 5, 4, 8, 7, 11, 11, 17, 16, 25, 24, 35, 35, 50, 49, 70, 69, 94, 96, 129, 129, 172, 174, 227, 232, 298, 303, 389, 396, 498, 513, 639, 655, 814, 834, 1025, 1057, 1287, 1326, 1610, 1657, 1995, 2063, 2469, 2548, 3039, 3138, 3720, 3851, 4539, 4696, 5523

Offset

1,2

Links

Table of n, a(n) for n=1..56.

Max Alekseyev, Proof of the conjecture

Formula

Conjecture: G.f. is x/(1-x)+((1-x^12)/((1-x^2)*(1-x^5)))*(1/Product_{k>0} (1-x^(prime(k)+1))), i.e. a(n) = 1 + number of partitions of n into parts of the form p+1, p a prime, excluding 12 and including 2 and 5. Added May 28 2008: The conjecture is correct! See the link. - Max Alekseyev

Maple

A139807 := proc(n) local g, i, p ; g := (1-x^12)/(1-x^2)/(1-x^5) ; for i from 1 do p := ithprime(i) ; if p > n then break ; fi ; g := taylor(g/(1-x^(p+1)), x=0, n+1) ; od: coeftayl( g+1/(1-x), x=0, n) ; end: seq(A139807(n), n=1..80) ; # R. J. Mathar, May 29 2008

Crossrefs

Cf. A034891.

Sequence in context: A351004 A114328 A097366 * A308465 A276119 A167755

Adjacent sequences: A139804 A139805 A139806 * A139808 A139809 A139810

Keyword

nonn

Author

Vladeta Jovovic, May 23 2008

Extensions

More terms from R. J. Mathar, May 29 2008

Status

approved