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A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).

%I A333910 #16 Jun 28 2022 01:46:16
%S A333910 1,3,7,10,17,18,19,20,21,22,27,30,31,36,40,44,45,46,50,51,55,57,58,60,
%T A333910 66,67,70,71,72,73,79,80,88,89,92,93,94,97,99,100,103,106,115,116,118,
%U A333910 119,120,126,127,132,133,138,140,144,145,150,154,160,162,163,165
%N A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).
%H A333910 Amiram Eldar, <a href="/A333910/b333910.txt">Table of n, a(n) for n = 1..10000</a>
%H A333910 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 A333910 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 A333910 1 is a term since psi(1) = 1 = 0^2 + 1^2.
%t A333910 psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[200], SquaresR[2, psi[#]] > 0 &]
%o A333910 (Python)
%o A333910 from itertools import count, islice
%o A333910 from collections import Counter
%o A333910 from sympy import factorint
%o A333910 def A333910_gen(): # generator of terms
%o A333910     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint(1+p))+Counter({p:e-1}) for p ,e in factorint(n).items()),start=Counter()).items()),count(1))
%o A333910 A333910_list = list(islice(A333910_gen(),30)) # _Chai Wah Wu_, Jun 27 2022
%Y A333910 Cf. A001481, A001615, A079545, A333909, A333911.
%K A333910 nonn
%O A333910 1,2
%A A333910 _Amiram Eldar_, Apr 09 2020