A349322 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).

1, 3, 2, 4, 2, 5, 2, 12, 11, 5, 2, 18, 2, 5, 4, 13, 2, 32, 2, 7, 4, 5, 2, 50, 3, 5, 12, 7, 2, 9, 2, 14, 4, 5, 4, 81, 2, 5, 4, 55, 2, 9, 2, 7, 14, 5, 2, 52, 3, 7, 4, 7, 2, 34, 4, 71, 4, 5, 2, 83, 2, 5, 14, 15, 4, 9, 2, 7, 4, 9, 2, 185, 2, 5, 6, 7, 4, 9, 2, 136, 13, 5, 2, 107

Offset

1,2

Comments

For each divisor d of n, add d if d is refactorable (i.e., if the number of divisors of d divides d), otherwise add 1. For example, the divisors of 8 are 1,2,4,8 and the refactorable divisors of 8 are 1,2,8. The sum is then a(8) = 1 + 2 + 1 + 8 = 12.

Inverse Möbius transform of n^c(n), where c = A336040. - Wesley Ivan Hurt, Jun 29 2024

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Eric Weisstein's World of Mathematics, Refactorable Number

Formula

a(n) = A335182(n) + A349658(n). - Antti Karttunen, Nov 24 2021

a(p) = 2 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

Mathematica

a[n_] := DivisorSum[n, If[Divisible[#, DivisorSigma[0, #]], #, 1] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

Prog

(PARI) isrf(n) = n%numdiv(n)==0; \\ A336040
a(n) = sumdiv(n, d, if (isrf(d), d, 1)); \\ Michel Marcus, Nov 16 2021

Crossrefs

Cf. A033950, A208251, A335182, A335665, A336040, A336041, A349658.

Sequence in context: A294111 A372715 A373556 * A193295 A356275 A256918

Adjacent sequences: A349319 A349320 A349321 * A349323 A349324 A349325

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 14 2021

Status

approved