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A373129
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a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( n/gcd(x_1, x_2, n) ).
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1
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1, 10, 33, 94, 145, 330, 385, 814, 969, 1450, 1441, 3102, 2353, 3850, 4785, 6766, 5185, 9690, 7201, 13630, 12705, 14410, 12673, 26862, 18745, 23530, 26889, 36190, 25201, 47850, 30721, 55150, 47553, 51850, 55825, 91086, 51985, 72010, 77649, 118030, 70561, 127050
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2 ).
a(n) = Sum_{d|n} J_2(d) * sigma(d), where the Jordan totient function J_2(n) = A007434(n).
Multiplicative with a(p^e) = (p^(3*e+2)*(p+1) - p^(2*e)*(p^2+p+1) + p)/(p^3-1).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.41347980562116709395... . (End)
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MATHEMATICA
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f[p_, e_] := (p^(3*e + 2)*(p+1) - p^(2*e)*(p^2+p+1) + p)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)
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PROG
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(PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=2, m=1) = sumdiv(n, d, J(d, k)*sigma(d^m));
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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