A335182 Sum of the refactorable divisors of n.

1, 3, 1, 3, 1, 3, 1, 11, 10, 3, 1, 15, 1, 3, 1, 11, 1, 30, 1, 3, 1, 3, 1, 47, 1, 3, 10, 3, 1, 3, 1, 11, 1, 3, 1, 78, 1, 3, 1, 51, 1, 3, 1, 3, 10, 3, 1, 47, 1, 3, 1, 3, 1, 30, 1, 67, 1, 3, 1, 75, 1, 3, 10, 11, 1, 3, 1, 3, 1, 3, 1, 182, 1, 3, 1, 3, 1, 3, 1, 131, 10, 3, 1, 99

Offset

1,2

Links

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

Eric Weisstein's World of Mathematics, Refactorable Number

Formula

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

a(n) = Sum_{d|n} d * (1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).

a(n) = A349322(n) - A349658(n). - Antti Karttunen, Nov 24 2021

a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

Example

a(6) = 3; The divisors of 6 are {1,2,3,6}. 1 and 2 are refactorable since d(1) = 1|1 and d(2) = 2|2, so a(6) = 1 + 2 = 3.
a(7) = 1; The divisors of 7 are {1,7} and 1 is the only refactorable divisor of 7. So a(7) = 1.
a(8) = 11; The divisors of 8 are {1,2,4,8}. 1, 2 and 8 are refactorable since d(1) = 1|1, d(2) = 2|2 and d(8) = 4|8, so a(8) = 1 + 2 + 8 = 11.
a(9) = 10; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9, so a(9) = 1 + 9 = 10.

Mathematica

a[n_] := DivisorSum[n, # &, Divisible[#, DivisorSigma[0, #]] &]; Array[a, 80] (* Amiram Eldar, Nov 24 2021 *)

Prog

(PARI) isr(n) = n%numdiv(n)==0; \\ A033950
a(n) = sumdiv(n, d, if (isr(d), d)); \\ Michel Marcus, Jul 20 2020

Crossrefs

Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A336041 (number of refactorable divisors), A335665 (their product).

Difference of A349322 and A349658.

Sequence in context: A319137 A074122 A372834 * A319992 A255894 A135023

Adjacent sequences: A335179 A335180 A335181 * A335183 A335184 A335185

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jul 17 2020.

Status

approved