A272405 Numbers n such that sum of the divisors of n is not of the form x^2 + y^2 + z^2 where x, y, z are integers.

4, 8, 12, 16, 18, 24, 25, 32, 38, 48, 59, 64, 75, 91, 96, 99, 114, 125, 128, 130, 135, 158, 166, 169, 177, 192, 196, 203, 205, 209, 221, 239, 242, 251, 256, 268, 273, 283, 290, 315, 324, 347, 358, 365, 367, 375, 378, 379, 384, 387, 390, 392, 403, 422, 423, 427, 443, 445, 460, 474, 476, 493

Offset

1,1

Comments

Numbers n such that sum of the positive divisors of n is the sum of 4 but no fewer nonzero squares.

Prime terms of this sequence are 59, 239, 251, 283, 347, 367, 379, 443, 571, ...

A006532 is a subsequence of complement of this sequence.

Pollack (2011) proved that the complementary sequence has asymptotic density 7/8. Therefore the asymptotic density of this sequence is 1/8. - Amiram Eldar, Apr 09 2020

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Paul Pollack, Values of the Euler and Carmichael functions which are sums of three squares, Integers, Vol. 11 (2011), pp. 145-161.

Formula

{n: A000203(n) in A004215}. - R. J. Mathar, May 02 2016

Example

1 is not a term since sigma(1) = 1 = 0^2 + 0^2 + 1^2 is the sum of 3 squares.
4 is a term since sigma(4) = 7 is not the sum of 3 squares.

Mathematica

Select[Range@ 500, ! SquaresR[3, DivisorSigma[1, #]] > 0 &] (* Michael De Vlieger, Apr 29 2016 *)

Prog

(PARI) isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA004215(sigma(n)), print1(n, ", ")));
(Python)
from itertools import count, islice
from sympy import divisor_sigma
def A272405_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~(s:=int(divisor_sigma(n)))&s-1).bit_length())&1 and (s>>m)&7==7, count(max(startvalue, 1)))
A272405_list = list(islice(A272405_gen(), 30)) # Chai Wah Wu, Jul 09 2022

Crossrefs

Cf. A000203, A004215, A006532.

Sequence in context: A068306 A311118 A348705 * A311119 A321177 A352490

Adjacent sequences: A272402 A272403 A272404 * A272406 A272407 A272408

Keyword

nonn,easy

Author

Altug Alkan, Apr 29 2016

Status

approved