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%I #15 Jun 28 2022 01:45:43
%S 1,3,7,9,10,17,19,21,22,27,30,31,40,46,51,52,55,57,58,63,66,67,70,71,
%T 73,79,81,88,89,90,93,94,97,103,106,115,118,119,120,127,133,138,145,
%U 153,154,156,163,165,170,171,174,179,184,189,190,193,198,199,201,202
%N Numbers k such that sigma(k) is the sum of 2 squares, where sigma is the sum of divisors function (A000203).
%H Amiram Eldar, <a href="/A333911/b333911.txt">Table of n, a(n) for n = 1..10000</a>
%H William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, <a href="https://doi.org/10.1093/qmath/hah039">Values of arithmetical functions equal to a sum of two squares</a>, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, <a href="https://faculty.missouri.edu/~bankswd/papers/2005_arith_functs_sum_two_squares.pdf">alternative link</a>.
%F c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).
%e 1 is a term since sigma(1) = 1 = 0^2 + 1^2.
%t Select[Range[200], SquaresR[2, DivisorSigma[1, #]] > 0 &]
%o (Python)
%o from itertools import count, islice
%o from collections import Counter
%o from sympy import factorint
%o def A333911_gen(): # generator of terms
%o return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint((p**(e+1)-1)//(p-1))) for p, e in factorint(n).items()),start=Counter()).items()),count(1))
%o A333911_list = list(islice(A333911_gen(),30)) # _Chai Wah Wu_, Jun 27 2022
%Y Cf. A000203, A001481, A079545, A272405, A333909, A333910.
%K nonn
%O 1,2
%A _Amiram Eldar_, Apr 09 2020
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