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 A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5. 2
 1, 35, 251, 1132, 3149, 8785, 16855, 36272, 61065, 110215, 161171, 284132, 371461, 589925, 790399, 1160896, 1420145, 2137275, 2476459, 3564668, 4230605, 5640985, 6436871, 9104272, 9841225, 13001135, 14839443, 19079860, 20511989, 27663965, 28630111, 37149440, 40453921 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities. FORMULA a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^2. a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(d), where mu is the Moebius function A008683. From Amiram Eldar, May 21 2024: (Start) Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1). Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s). Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End) MATHEMATICA f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *) PROG (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 3)); CROSSREFS Cf. A069097, A360428, A368743, A372926. Cf. A001158, A008683. Cf. A013662, A013664, A088246. Sequence in context: A255584 A113941 A067238 * A267022 A145014 A090646 Adjacent sequences: A372924 A372925 A372926 * A372928 A372929 A372930 KEYWORD nonn,mult AUTHOR Seiichi Manyama, May 17 2024 STATUS approved

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Last modified August 3 05:44 EDT 2024. Contains 374875 sequences. (Running on oeis4.)