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A284118 Sum of nonprime squarefree divisors of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A271498 A317940 A318674 * A165725 A214685 A327670

Adjacent sequences:  A284115 A284116 A284117 * A284119 A284120 A284121

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 20 2017

STATUS

approved

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Last modified September 17 05:34 EDT 2021. Contains 347478 sequences. (Running on oeis4.)