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A285309 Sum of nonsquare divisors of n. 4
0, 2, 3, 2, 5, 11, 7, 10, 3, 17, 11, 23, 13, 23, 23, 10, 17, 29, 19, 37, 31, 35, 23, 55, 5, 41, 30, 51, 29, 71, 31, 42, 47, 53, 47, 41, 37, 59, 55, 85, 41, 95, 43, 79, 68, 71, 47, 103, 7, 67, 71, 93, 53, 110, 71, 115, 79, 89, 59, 163, 61, 95, 94, 42, 83, 143, 67, 121, 95, 143, 71, 145, 73, 113, 98 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to sums of divisors

FORMULA

G.f.: Sum_{k>=1} A000037(k)*x^A000037(k)/(1 - x^A000037(k)).

a(n) = A000203(n) - A035316(n).

a(A005117(n)) = A000203(A005117(n)) - 1.

a(p^(2*k-1)) = a(p^(2*k)) = p*(p^(2*k) - 1)/(p^2 - 1) for p is a prime and k >= 1.

EXAMPLE

a(6) = 11 because 6 has 4 divisors {1, 2, 3, 6} among which 3 are nonsquares {2, 3, 6} therefore 2 + 3 + 6 = 11.

MATHEMATICA

Table[DivisorSum[n, # &, Mod[DivisorSigma[0, #], 2] == 0 &], {n, 1, 75}]

nmax = 75; Rest[CoefficientList[Series[Sum[(k + Floor[1/2 + Sqrt[k]]) x^(k + Floor[1/2 + Sqrt[k]])/(1 - x^(k + Floor[1/2 + Sqrt[k]])), {k, 1, nmax}], {x, 0, nmax}], x]]

Array[DivisorSum[#, # &, ! IntegerQ@ Sqrt@ # &] &, 75] (* Michael De Vlieger, Nov 23 2017 *)

PROG

(PARI) a(n) = sumdiv(n, d, if (!issquare(d), d)); \\ Michel Marcus, Apr 17 2017

(Python)

import gmpy

from sympy import divisors

def a(n): return sum([d for d in divisors(n) if gmpy.is_square(d)==0]) # Indranil Ghosh, Apr 18 2017

CROSSREFS

Cf. A000037, A000203, A005117, A035316, A056595, A087153.

Sequence in context: A082050 A183098 A183101 * A250096 A162687 A010242

Adjacent sequences:  A285306 A285307 A285308 * A285310 A285311 A285312

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 16 2017

STATUS

approved

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Last modified January 21 18:42 EST 2021. Contains 340352 sequences. (Running on oeis4.)