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A349322
a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).
5
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Nov 14 2021
STATUS
approved