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Catalina Calderón, Jose Maria Grau, A. Oller-Marcén, and László Tóth, <a href="http://dx.doi.org/10.5486/PMD.2015.7098">Counting invertible sums of squares modulo n and a new generalization of Euler's totient function</a>, Publicationes Mathematicae-Debrecen, Vol. 87 (1-2) (2015), pp. 133-145; <a href="https://arxiv.org/abs/1403.7878">arXiv preprint</a>, arXiv:1403.7878 [math.NT], 2014.

C. Catalina Calderón, J. M. Jose Maria Grau, and A. Oller-Marcén, and László Tóth, <a href="http://arxivdx.doi.org/abs10.5486/1403PMD.2015.78787098">Counting invertible sums of squares modulo n and a new generalization of Euler 's totient function</a>, Publicationes Mathematicae-Debrecen, Vol. 87 (2015), pp. 133-145; <a href="https://arxiv.org/abs/1403.7878">arXiv preprint</a>, arXiv:1403.7878 [math.NT], 2014.

C. Calderón, J. M. Grau, and A. Oller-Marcén, <a href="http://publi.math.unideb.hu/contents.php?szam=87">Counting invertible sums of squares modulo n and a new generalization of Euler totient function</a>, Publicationes mathematicae 87(1-2) (2015), 133-145.

From Amiram Eldar, Feb 13 2024: (Start)

Dirichlet g.f.: zeta(s-6) * (1 - 1/2^(s-5)) * Product_{p prime > 2} (1 - 1/p^(s-5) - (-1)^(3*(p-1)/2)*(p-1)/p^(s-2)).

Sum_{k=1..n} a(k) = c * n^7 + O(n^6 * log(n)), where c = (3/28) * Product_{p prime == 1 (mod 4)} (1 - 1/p^2 - 1/p^4 + 1/p^5) * Product_{p prime == 3 (mod 4)} (1 - 1/p^2 + 1/p^4 - 1/p^5) = 0.08756841635... (Calderón et al., 2015). (End)

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f[p_, e_] := (p - 1)*p^(6*e - 4)*(p^3 - (-1)^(3*(p - 1)/2)); f[2, e_] := 2^(6*e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 07 2023 *)

nonn,easy,mult

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