A344371 a(n) = Sum_{k=1..n} (-1)^(n-k) gcd(k,n).

1, 1, 3, 4, 5, 5, 7, 12, 9, 9, 11, 20, 13, 13, 15, 32, 17, 21, 19, 36, 21, 21, 23, 60, 25, 25, 27, 52, 29, 45, 31, 80, 33, 33, 35, 84, 37, 37, 39, 108, 41, 65, 43, 84, 45, 45, 47, 160, 49, 65, 51, 100, 53, 81, 55, 156, 57, 57, 59, 180, 61, 61, 63, 192, 65, 105

Offset: 1

Author

Max Alekseyev, May 16 2021

Keywords

• nonn
• easy

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001620, A090423, A106475, A199084, A306016, A344372, A344373.

Programs

PARI

a(n) = sum(k=1, n, (-1)^(n-k)*gcd(k,n)); \\ Michel Marcus, May 16 2021

Formula

a(n) = abs(A199084(n)).

a(2n+1) = 2n+1.

a(2n) = A344372(n) = 2*n - A106475(n-1).

Sum_{k=1..n} a(k) ~ (n^2/Pi^2) * (log(n) + 2*gamma - 1/2 - 4*log(2)/3 + Pi^2/4 - zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

Extensions

More terms from Felix Fröhlich, May 19 2021