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

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

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences related to divisors of numbers.

Formula

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.

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

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

Mathematica

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]

Prog

(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)); }

Crossrefs

Cf. A001694, A049417, A077609, A077610, A360723.

Similar sequences: A183097, A360721.

Sequence in context: A242404 A145295 A366990 * A091051 A183097 A353958

Adjacent sequences: A360719 A360720 A360721 * A360723 A360724 A360725

Keyword

nonn,mult,easy

Author

Amiram Eldar, Feb 18 2023

Status

approved