cp's OEIS Frontend

A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

%I A360722 #11 Feb 26 2023 03:27:31
%S A360722 1,1,1,5,1,1,1,13,10,1,1,5,1,1,1,17,1,10,1,5,1,1,1,13,26,1,37,5,1,1,1,
%T A360722 49,1,1,1,50,1,1,1,13,1,1,1,5,10,1,1,17,50,26,1,5,1,37,1,13,1,1,1,5,1,
%U A360722 1,10,85,1,1,1,5,1,1,1,130,1,1,26,5,1,1,1,17
%N A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).
%H A360722 Amiram Eldar, <a href="/A360722/b360722.txt">Table of n, a(n) for n = 1..10000</a>
%H A360722 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.
%F A360722 Multiplicative with a(p^e) = f(p, e) if e is even, and f(p, e) - p is e is odd, where f(p, e) = Product{k>=1, e_k=1} (p^(2^k) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
%F A360722 a(n) <= A049417(n), with equality if and only if n is a square.
%F A360722 a(n) <= A183097(n), with equality if and only if n is not in A360723.
%t A360722 f[p_, e_] := Times @@ (p^(2^(-1 + Flatten @ Position[Reverse@IntegerDigits[e, 2], _?(# == 1 &)])) + 1) - If[OddQ[e], p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360722 (PARI) a(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k))+1, 1)) - if(f[i, 2]%2, f[i, 1], 0));}
%Y A360722 Cf. A001694, A049417, A077609, A077610, A360723.
%Y A360722 Similar sequences: A183097, A360721.
%K A360722 nonn,mult,easy
%O A360722 1,4
%A A360722 _Amiram Eldar_, Feb 18 2023