A325310 a(n) = A001511(A325315(n)), except when A325315(n) = 0, then a(n) = 0.

1, 1, 2, 1, 3, 0, 2, 1, 1, 2, 2, 3, 3, 3, 2, 1, 5, 1, 2, 2, 2, 4, 2, 3, 1, 2, 2, 0, 3, 3, 2, 1, 2, 2, 2, 1, 3, 5, 2, 2, 4, 3, 2, 3, 3, 3, 2, 3, 1, 1, 2, 2, 3, 3, 2, 4, 2, 2, 2, 4, 3, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 1, 4, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 6, 2, 4, 2, 3, 4, 2, 2, 5, 2, 3, 2, 3, 6, 1, 2, 1, 3, 3, 2, 2, 2

Offset: 1

Author

Antti Karttunen, Apr 21 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A000396, A001511, A028982 (gives the positions of 1's), A048250, A162296, A228058, A325313, A325314, A325315, A325378, A325379.

Programs

Mathematica

Array[If[# == 0, 0, IntegerExponent[2 #, 2]] &[BitXor @@ Abs[#1 - Map[Total, {#3, Complement[#2, #3]}]]] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 105] (* Michael De Vlieger, Apr 21 2019 *)

PARI

A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325315(n) = bitxor(abs(A325313(n)),abs(A325314(n)));
A325310(n) = A001511ext(A325315(n));

Formula

If A325315(n) = 0, then a(n) = 0, otherwise a(n) = A001511(A325315(n)).

a(A228058(n)) = A001511(abs(A325379(n))), assuming there are no odd perfect numbers, in which case a(A228058(n)) >= 3 for all n.