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