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A272405
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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.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range@ 500, ! SquaresR[3, DivisorSigma[1, #]] > 0 &] (* Michael De Vlieger, Apr 29 2016 *)
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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