0,3

Comment from Joerg Arndt, Jun 08 2009: Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part."

The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5) - 1 = 6, a(7) = A000041(7) - 3 = 12. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 17 2003

Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 08 2009]

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 27 2008

Comment from Joerg Arndt, Jun 08 2009: Sequence has g.f. 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n.

Gary Adamson's comment is equivalent to the formula a(n) = Sum_{d|n} p(d-1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d-1)*x^d/(1-x^d), - N. J. A. Sloane, Jun 08 2009

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-0 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 14 2007

Cf. A083711, A018783, A137587.

a000041, A051731 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 08 2009]

Sequence in context: A161715 A164523 A033159 this_sequence A127524 A117086 A081026

Adjacent sequences: A083707 A083708 A083709 this_sequence A083711 A083712 A083713

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Jun 16 2003

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 17 2003