A296134 Number of twice-factorizations of n of type (R,Q,R).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset

1,4

Comments

a(n) is the number of ways to choose a strict integer partition of a divisor of A052409(n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

Formula

From Antti Karttunen, Jul 31 2018: (Start)

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000009(d).

a(n) = A047966(A052409(n)). (End)

Example

The a(16) = 4 twice-factorizations: (2)*(2*2*2), (2*2*2*2), (4*4), (16).

Mathematica

Table[DivisorSum[GCD@@FactorInteger[n][[All, 2]], PartitionsQ], {n, 100}]

Prog

(PARI)
A000009(n, k=(n-!(n%2))) = if(!n, 1, my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k, k)); k -= 2); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A296134(n) = if(1==n, n, sumdiv(A052409(n), d, A000009(d))); \\ Antti Karttunen, Jul 29 2018

Crossrefs

Cf. A000005, A000009, A001055, A047966, A047968, A052409, A052410, A089723, A281113, A295923, A295924, A295931, A295935, A296133.

Sequence in context: A267115 A328919 A277647 * A306694 A158052 A253641

Adjacent sequences: A296131 A296132 A296133 * A296135 A296136 A296137

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Extensions

More terms from Antti Karttunen, Jul 29 2018

Status

approved