A330319 a(n) = Sum_{i=1..n} phi(i)*phi(i+1), where phi(n) = A000010(n) is Euler's totient function.

1, 3, 7, 15, 23, 35, 59, 83, 107, 147, 187, 235, 307, 355, 419, 547, 643, 751, 895, 991, 1111, 1331, 1507, 1667, 1907, 2123, 2339, 2675, 2899, 3139, 3619, 3939, 4259, 4643, 4931, 5363, 6011, 6443, 6827, 7467, 7947, 8451, 9291, 9771, 10299, 11311, 12047, 12719, 13559, 14199, 14967, 16215, 17151, 17871, 18831

Offset: 1

Author

N. J. A. Sloane, Dec 11 2019

Keywords

• nonn

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 32.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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.
• L. Mirsky, Summation formula involving arithmetic functions, Duke Mathematical Journal, Vol. 16, No. 2 (1949), pp. 261-272.

Crossrefs

Cf. A000010, A065474, A335131.

Partial sums of A083542.

Programs

Mathematica

phi = EulerPhi[Range[56]]; Accumulate[Most[phi] * Rest[phi]] (* Amiram Eldar, Mar 05 2020 *)

PARI

a(n) = sum(i=1, n, eulerphi(i)*eulerphi(i+1)); \\ Michel Marcus, Mar 05 2020

Formula

a(n) ~ (c/3) * n^3 + O(n^2*log(n)^2), where c = Product_{p prime}(1 - 2/p^2) (A065474). - Amiram Eldar, Mar 05 2020