A011755 a(n) = Sum_{k=1..n} k*phi(k).

1, 3, 9, 17, 37, 49, 91, 123, 177, 217, 327, 375, 531, 615, 735, 863, 1135, 1243, 1585, 1745, 1997, 2217, 2723, 2915, 3415, 3727, 4213, 4549, 5361, 5601, 6531, 7043, 7703, 8247, 9087, 9519, 10851, 11535, 12471, 13111, 14751, 15255, 17061, 17941, 19021, 20033

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) = Sum_{(x,y): 1<=x<=y<=n, 1=gcd(x,y)} y. Sum_{(x,y): 1<=x<=y<=n, 1=gcd(x,y)} x = (a(n)+1)/2. - Vladeta Jovovic, Jan 02 2003

Equals row sums of triangle A110663. Example: a(4) = 17 = (6 + 5 + 4 + 2), where row 4 of triangle A110663 = (6, 5, 4, 2). - Gary W. Adamson, Jul 26 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..2001 from Indranil Ghosh)

Crossrefs

Partial sums of A002618.

Cf. A000010, A002088, A110663, A319087.

Programs

Mathematica

Accumulate[Table[k*EulerPhi[k], {k, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

PARI

a(n) = sum(k=1, n, k*eulerphi(k)); \\ Michel Marcus, Feb 13 2017

Python

from sympy import totient
def A011755(n): return sum(k*totient(k) for k in range(1,n+1)) # Chai Wah Wu, Feb 21 2023

Formula

Asymptotically: a(n) ~ C*n^3 with C=0.20264.... - Benoit Cloitre, Jan 14 2002

Asymptotically: a(n) ~ (2/Pi^2)*n^3. - Vladeta Jovovic, Jan 02 2003

a(n) = Sum_{k=1..n} phi(k^2). - Vaclav Kotesovec, May 08 2024