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A341772
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a(n) = Sum_{d|n} phi(d) * J_2(n/d).
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1
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1, 4, 10, 17, 28, 40, 54, 70, 94, 112, 130, 170, 180, 216, 280, 284, 304, 376, 378, 476, 540, 520, 550, 700, 716, 720, 858, 918, 868, 1120, 990, 1144, 1300, 1216, 1512, 1598, 1404, 1512, 1800, 1960, 1720, 2160, 1890, 2210, 2632, 2200, 2254, 2840, 2682, 2864, 3040, 3060, 2860, 3432, 3640
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OFFSET
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1,2
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COMMENTS
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Dirichlet convolution of Euler totient function phi (A000010) with Jordan function J_2 (A007434).
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.
a(n) = Sum_{k=1..n} J_2(gcd(n,k)).
a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).
a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).
a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).
a(n) = Sum_{d|n} d^2 * A007431(n/d).
a(n) = Sum_{d|n} mu(n/d) * A069097(d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - Vaclav Kotesovec, Feb 20 2021
a(n) = Sum_{k=1..n} J_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = Sum_{1 <= i, j <= n} phi(gcd(i, j, n)). - Peter Bala, Jan 21 2024
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MATHEMATICA
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Jordan2[n_] := Sum[MoebiusMu[n/d] d^2, {d, Divisors[n]}]; a[n_] := Sum[EulerPhi[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 55}]
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PROG
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(PARI) J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434
a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ Michel Marcus, Feb 20 2021
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CROSSREFS
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Cf. A000010, A000203, A001615, A002618, A007427, A007431, A007434, A008683, A023900, A029935, A064987, A069097, A321322, A322577, A338164.
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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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