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A092931 Number of ways of factorizing n into parts whose sum divides n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
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
REFERENCES
Amarnath Murthy, "Generalization of partition function, introducing Smarandache Factor partition", Smarandache Notions Journal, Vol. 11, 1-2-3, 2000.
LINKS
Richard J Mathar, Maple program
EXAMPLE
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.
CROSSREFS
Sequence in context: A374381 A177121 A329314 * A147300 A346481 A240228
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Mar 20 2004
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
STATUS
approved

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Last modified July 17 05:50 EDT 2024. Contains 374360 sequences. (Running on oeis4.)