A333315 a(n) = Sum_{k=1..n} phi(prime(k)-1), where phi is the Euler totient function (A000005).

1, 2, 4, 6, 10, 14, 22, 28, 38, 50, 58, 70, 86, 98, 120, 144, 172, 188, 208, 232, 256, 280, 320, 360, 392, 432, 464, 516, 552, 600, 636, 684, 748, 792, 864, 904, 952, 1006, 1088, 1172, 1260, 1308, 1380, 1444, 1528, 1588, 1636, 1708, 1820, 1892, 2004, 2100, 2164

Offset: 1

Author

Amiram Eldar, Mar 14 2020

Keywords

• nonn

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 30.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• S. S. Pillai, On the sum function connected with primitive roots, Proceedings of the Indian Academy of Sciences - Section A, Vol. 13 (1941), pp. 526-529, alternative link.
• Eric Weisstein's World of Mathematics, Logarithmic Integral.
• Wikipedia, Logarithmic integral function.

Crossrefs

Partial sums of A008330.

Cf. A000005, A005596.

Programs

Mathematica

Accumulate @ EulerPhi[Select[Range[300], PrimeQ] - 1]

PARI

a(n) = sum(k=1, n, eulerphi(prime(k)-1)); \\ Michel Marcus, Mar 15 2020

Formula

a(n) = Sum_{k=1..n} A008330(k).

a(n) ~ A * Li(n^2), where A is Artin's constant (A005596), and Li(x) is the logarithmic integral function.