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

Author

Gus Wiseman, Dec 05 2017

Keywords

• nonn

Comments

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

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

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

Programs

Mathematica

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

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

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)

Extensions

More terms from Antti Karttunen, Jul 29 2018