login
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
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
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
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Apr 29 2016
STATUS
approved