A092931 Number of ways of factorizing n into parts whose sum divides n.

0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2

Offset: 1

Author

Amarnath Murthy, Mar 20 2004

Keywords

• nonn

Comments

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, Mar 21 2006

Examples

			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.
		

References

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

Links

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

Crossrefs

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

Extensions

More terms from Jonathan Vos Post, Mar 21 2006

More terms from Franklin T. Adams-Watters, Jun 12 2006

a(100) corrected by N. J. A. Sloane, Nov 23 2007