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

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

Author

Ilya Gutkovskiy, Apr 11 2019

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Dedekind Function

Crossrefs

Cf. A001615, A008683, A065093, A307309.

Programs

Mathematica

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}]

Formula

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