A284118 Sum of nonprime squarefree divisors of n.

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 7, 1, 15, 16, 1, 1, 7, 1, 11, 22, 23, 1, 7, 1, 27, 1, 15, 1, 62, 1, 1, 34, 35, 36, 7, 1, 39, 40, 11, 1, 84, 1, 23, 16, 47, 1, 7, 1, 11, 52, 27, 1, 7, 56, 15, 58, 59, 1, 62, 1, 63, 22, 1, 66, 128, 1, 35, 70, 130, 1, 7, 1, 75, 16, 39, 78, 150, 1, 11, 1, 83, 1, 84, 86, 87, 88, 23, 1, 62

Offset: 1

Author

Ilya Gutkovskiy, Mar 20 2017

Keywords

• nonn

Examples

			a(30) = 62 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} therefore 1 + 6 + 10 + 15 + 30 = 62.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000469, A001221, A008472, A008683, A048250, A259936.

Programs

Mathematica

nmax = 90; Rest[CoefficientList[Series[x/(1 - x) + Sum[Sign[PrimeNu[k] - 1] MoebiusMu[k]^2 k x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], #1 == 1 || (SquareFreeQ[#1] && PrimeNu[#1] > 1) &]], {n, 90}]

PARI

Vec((x/(1 - x)) + sum(k=2, 90, sign(omega(k) - 1) * moebius(k)^2 * k * x^k/(1 - x^k)) + O(x^91)) \\ Indranil Ghosh, Mar 21 2017

Python

from sympy import divisors
from sympy.ntheory.factor_ import core
def a(n): return sum([i for i in divisors(n) if core(i)==i and isprime(i)==0]) # Indranil Ghosh, Mar 21 2017

Formula

G.f.: x/(1 - x) + Sum_{k>=2} sgn(omega(k)-1)*mu(k)^2*k*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221) and mu(k) is the Moebius function (A008683).

a(n) = Sum_{d|n, d nonprime squarefree} d.

a(n) = 1 if n is a prime power.

a(n) = A048250(n) - A008472(n). - Amiram Eldar, Jun 05 2025