A065093 Convolution of A000010 with itself.

1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120, 124, 188, 196, 276, 272, 404, 380, 544, 532, 716, 668, 968, 860, 1184, 1120, 1472, 1332, 1896, 1624, 2204, 2036, 2656, 2352, 3284, 2752, 3684, 3356, 4324, 3744, 5192, 4312, 5720, 5180, 6540, 5628, 7768, 6388, 8476

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.

Formula

a(n) = Sum_{k=1..n} phi(k)*phi(n+1-k), where phi is Euler totient function (A000010).

G.f.: (1/x)*(Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2)^2. - Ilya Gutkovskiy, Jan 31 2017

a(n) ~ (n^3/6) * c * Product_{primes p|n+1} ((p^3-2*p+1)/(p*(p^2-2))), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Ingham, 1927). - Amiram Eldar, Jul 13 2024

Mathematica

Table[Sum[EulerPhi[j]*EulerPhi[n-j], {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Aug 18 2021 *)

Prog

(PARI) { for (n=1, 1000, a=sum(k=1, n, eulerphi(k)*eulerphi(n+1-k)); write("b065093.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

Crossrefs

Cf. A000010, A000385, A055507, A065474, A330319.

Column k=2 of A340995.

Sequence in context: A257548 A186413 A080084 * A361270 A360671 A168470

Adjacent sequences: A065090 A065091 A065092 * A065094 A065095 A065096

Keyword

easy,nonn

Author

Vladeta Jovovic, Nov 11 2001

Status

approved