A285309 Sum of nonsquare divisors of n.

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

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: A183098 A183101 A343300 * A250096 A345302 A357183

Adjacent sequences: A285306 A285307 A285308 * A285310 A285311 A285312

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 16 2017

Status

approved