cp's OEIS Frontend

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.

%I A272405 #24 Jul 09 2022 11:06:38
%S A272405 4,8,12,16,18,24,25,32,38,48,59,64,75,91,96,99,114,125,128,130,135,
%T A272405 158,166,169,177,192,196,203,205,209,221,239,242,251,256,268,273,283,
%U A272405 290,315,324,347,358,365,367,375,378,379,384,387,390,392,403,422,423,427,443,445,460,474,476,493
%N 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.
%C A272405 Numbers n such that sum of the positive divisors of n is the sum of 4 but no fewer nonzero squares.
%C A272405 Prime terms of this sequence are 59, 239, 251, 283, 347, 367, 379, 443, 571, ...
%C A272405 A006532 is a subsequence of complement of this sequence.
%C A272405 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
%H A272405 Amiram Eldar, <a href="/A272405/b272405.txt">Table of n, a(n) for n = 1..10000</a>
%H A272405 Paul Pollack, <a href="https://www.emis.de/journals/INTEGERS/papers/l13/l13.Abstract.html">Values of the Euler and Carmichael functions which are sums of three squares</a>, Integers, Vol. 11 (2011), pp. 145-161.
%F A272405 {n: A000203(n) in A004215}. - _R. J. Mathar_, May 02 2016
%e A272405 1 is not a term since sigma(1) = 1 = 0^2 + 0^2 + 1^2 is the sum of 3 squares.
%e A272405 4 is a term since sigma(4) = 7 is not the sum of 3 squares.
%t A272405 Select[Range@ 500, ! SquaresR[3, DivisorSigma[1, #]] > 0 &] (* _Michael De Vlieger_, Apr 29 2016 *)
%o A272405 (PARI) isA004215(n) = {n\4^valuation(n, 4)%8==7}
%o A272405 lista(nn) = for(n=1, nn, if(isA004215(sigma(n)), print1(n, ", ")));
%o A272405 (Python)
%o A272405 from itertools import count, islice
%o A272405 from sympy import divisor_sigma
%o A272405 def A272405_gen(startvalue=1): # generator of terms >= startvalue
%o A272405     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)))
%o A272405 A272405_list = list(islice(A272405_gen(),30)) # _Chai Wah Wu_, Jul 09 2022
%Y A272405 Cf. A000203, A004215, A006532.
%K A272405 nonn,easy
%O A272405 1,1
%A A272405 _Altug Alkan_, Apr 29 2016