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A280184
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Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.
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10
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1, 16, 41, 136, 157, 656, 401, 1096, 1121, 2512, 1465, 5576, 2381, 6416, 6437, 8776, 5221, 17936, 7241, 21352, 16441, 23440, 12721, 44936, 19657, 38096, 30281, 54536, 25261, 102992, 30785, 70216, 60065, 83536, 62957, 152456, 52061, 115856, 97621, 172072, 70645, 263056, 81401, 199240, 175997, 203536, 106081, 359816, 137601, 314512
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{a|n, b|n, c|n, d|n} phi(a)*phi(b)*phi(c)*phi(d)/phi(lcm(a, b, c, d)), where phi is Euler totient function (cf. A000010).
Multiplicative with a(p^e) = 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.5010902655... . (End)
a(n) = Sum_{d divides n} J_4(d)/phi(d) = Sum_{1 <= i, j, k, l <= n} 1/phi(n/gcd(i,j,k,l,n)), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024
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MAPLE
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with(numtheory):
# define Jordan totient function J(r, n)
J(r, n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(4, d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024
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MATHEMATICA
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a[n_] := With[{dd = Divisors[n]}, Sum[Times @@ EulerPhi @ {x, y, z, t} / EulerPhi[LCM[x, y, z, t]], {x, dd}, {y, dd}, {z, dd}, {t, dd}]];
f[p_, e_] := 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)
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PROG
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(PARI) a(n) = sumdiv(n, x, sumdiv(n, y, sumdiv(n, z, sumdiv(n, t, eulerphi(x)*eulerphi(y)*eulerphi(z)*eulerphi(t)/eulerphi(lcm([x, y, z, t])))))); \\ Michel Marcus, Feb 26 2018
(PARI) a160891(n) = sumdiv(n, d, moebius(n/d)*d^4)/eulerphi(n);
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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