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a(n) is the sum of infinitary divisors of n that are powerful (A001694).
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%I #11 Feb 26 2023 03:27:31

%S 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 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 1,10,85,1,1,1,5,1,1,1,130,1,1,26,5,1,1,1,17

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

%H Amiram Eldar, <a href="/A360722/b360722.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.

%F 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 a(n) <= A049417(n), with equality if and only if n is a square.

%F a(n) <= A183097(n), with equality if and only if n is not in A360723.

%t 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 (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 Cf. A001694, A049417, A077609, A077610, A360723.

%Y Similar sequences: A183097, A360721.

%K nonn,mult,easy

%O 1,4

%A _Amiram Eldar_, Feb 18 2023