OFFSET
0,3
COMMENTS
Also number of ways to choose a divisor d|n and then a sequence of n/d divisors of d.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..6643
FORMULA
a(n) = Sum_{d|n} tau(n/d)^d for n > 0. - Andrew Howroyd, Aug 26 2018
G.f.: 1 + Sum_{k>=1} tau(k)*x^k/(1 - tau(k)*x^k). - Ilya Gutkovskiy, May 23 2019
a(n) = 3 <=> n is prime <=> n in { A000040 }. - Alois P. Heinz, May 23 2019
EXAMPLE
The a(6)=17 twice-constant partitions are:
((6)),
((3)(3)), ((33)),
((3)(111)), ((111)(3)),
((2)(2)(2)), ((222)),
((2)(2)(11)), ((2)(11)(2)), ((11)(2)(2)),
((2)(11)(11)), ((11)(2)(11)), ((11)(11)(2)),
((1)(1)(1)(1)(1)(1)), ((11)(11)(11)), ((111)(111)), ((111111)).
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
add(tau(n/d)^d, d=divisors(n)))
end:
seq(a(n), n=0..70); # Alois P. Heinz, Dec 20 2016
MATHEMATICA
nn=20; Table[DivisorSum[n, Power[DivisorSigma[0, #], n/#]&], {n, nn}]
PROG
(PARI) a(n)=if(n==0, 1, sumdiv(n, d, numdiv(n/d)^d)) \\ Andrew Howroyd, Aug 26 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 18 2016
STATUS
approved