[go: up one dir, main page]

login
A296134
Number of twice-factorizations of n of type (R,Q,R).
5
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).
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2017
EXTENSIONS
More terms from Antti Karttunen, Jul 29 2018
STATUS
approved