A307309 -id:A307309 - cp's OEIS frontend

A307308 Self-composition of the Euler totient function (A000010).

1, 2, 6, 15, 42, 106, 280, 702, 1778, 4398, 10910, 26678, 65172, 157656, 380524, 912846, 2185906, 5216588, 12433166, 29564544, 70189672, 166245574, 392909240, 926290066, 2178881218, 5114469170, 11985221654, 28049398284, 65588182636, 153277006212, 358073997608

Offset: 1

Ilya Gutkovskiy, Apr 02 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Totient Function

Cf. A000010, A008683, A010554, A065093, A307309.

g[x_] := g[x] = Sum[MoebiusMu[k] x^k/(1 - x^k)^2, {k, 1, 31}]; a[n_] := a[n] = SeriesCoefficient[g[g[x]], {x, 0, n}]; Table[a[n], {n, 31}]

G.f.: g(g(x)), where g(x) = Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2 is the g.f. of A000010.

A307502 Self-convolution of the Dedekind psi function (A001615).

Original entry on oeis.org

0, 1, 6, 17, 36, 64, 108, 172, 240, 340, 444, 612, 744, 980, 1164, 1504, 1704, 2172, 2388, 2964, 3288, 3968, 4272, 5272, 5520, 6624, 7104, 8276, 8640, 10404, 10572, 12480, 13032, 14988, 15300, 18204, 18048, 21004, 21636, 24616, 24648, 29036, 28452, 32768, 33552, 37488

Offset: 1

Ilya Gutkovskiy, Apr 11 2019

nonn

Eric Weisstein's World of Mathematics, Dedekind Function

Cf. A001615, A008683, A065093, A307309.

Rest[nmax = 46; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k)^2, {k, 1, nmax}]^2, {x, 0, nmax}], x]]
psi[n_] := psi[n] = Sum[MoebiusMu[n/d]^2 d, {d, Divisors @ n}]; a[n_] := a[n] = Sum[psi[k] psi[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 46}]

G.f.: (Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2)^2.
a(n) = Sum_{k=1..n-1} A001615(k)*A001615(n-k).
Conjecture: Sum_{k=1..n} a(k) ~ 75 * n^4 / (8 * Pi^4). - Vaclav Kotesovec, Aug 20 2025

cp's OEIS Frontend

A307308 Self-composition of the Euler totient function (A000010).

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