1,4

Most of the terms are 1. But there are infinitely many terms for which a(n) >1. Example: a(n^n) >= 2, two such factorizations being n^n and n*n*n... n times, e.g. a(27) = 2 from 27, 3*3*3.

For any prime p the only factorization of p is p, which sums to p, which divides p, hence a(p) = 1. For the square of any positive even number e = 2*k we have e^2 = (2*k)^2 = 4*k^2; since we can factor e^2 as (2*k)*(2*k) whose factors sum to 4*k and 4*k | 4*k^2, we have a((2*k)^2) >= 2. For any odd semiprime s = p*q, s in A046315, we have p+q is even, hence p+q cannot divide p*q, hence a(p*q) = 1. For any even semiprime s > 4, s in A100484, we have s = 2*p for an odd prime p, hence 2+p is odd an cannot divide either 2 nor p, so a(2*p) = 1. See also: A016742 Even squares: (2n)^2. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 21 2006

Amarnath Murthy, "Generalization of partition function, introducing Smarandache Factor partition", Smarandache Notions Journal, Vol. 11, 1-2-3, 2000.

Richard J Mathar, Table of n, a(n) for n = 1..10000

Richard J Mathar, Maple program

a(1) = 0. The only factorization of 1 is the empty multiset, whose sum is 0 and that does not divide 1.

a(16) = 4, the factorizations of 16 are 16, 8*2, 4*4, 4*2*2, 2*2*2*2. In four of them, all except 8*2, the sum of the parts divides 16.

a(30) = 2 because (besides 30 itself) we have 30 = 2 * 3 * 5 and 2 + 3 + 5 = 10 which divides 30.

a(100) = 3 from 100 = 5*20 = 10*10.

Cf. A000040, A016742, A046315, A100484, A092932.

Sequence in context: A043285 A146291 A086251 this_sequence A147300 A110503 A030556

Adjacent sequences: A092928 A092929 A092930 this_sequence A092932 A092933 A092934

nonn

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 20 2004

More terms from Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 21 2006

More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 12 2006

a(100) corrected by N. J. A. Sloane (njas(AT)research.att.com), Nov 23 2007