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

Author

Altug Alkan, Apr 29 2016

Keywords

• nonn
• easy

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

Examples

			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.
		

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.

Crossrefs

Cf. A000203, A004215, A006532.

Programs

Mathematica

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

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

Formula

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