A284118 - OEIS

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

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OFFSET

1,6

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

Index entries for sequences related to sums of divisors

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.

EXAMPLE

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.

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}]

PROG

(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

CROSSREFS

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

Sequence in context: A348281 A317940 A318674 * A165725 A214685 A327670

Adjacent sequences: A284115 A284116 A284117 * A284119 A284120 A284121

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 20 2017

STATUS

approved