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

Author

Wesley Ivan Hurt, Jul 17 2020

Keywords

• nonn
• easy

Comments

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of refactorable numbers (A336040). - Wesley Ivan Hurt, Jun 21 2024

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Eric Weisstein's World of Mathematics, Refactorable Number

Crossrefs

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

Difference of A349322 and A349658.

Programs

Mathematica

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

PARI

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

Formula

a(n) = Sum_{d|n} d * c(d), where c = 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