A115005 a(n) = (A114043(n) - 1)/2.

0, 3, 14, 43, 100, 209, 374, 641, 1020, 1553, 2246, 3197, 4372, 5911, 7778, 10037, 12728, 16043, 19862, 24467, 29728, 35777, 42626, 50625, 59520, 69675, 80966, 93627, 107568, 123345, 140458, 159673, 180664, 203651, 228590, 255857, 285116, 317363, 352058

Offset: 1

Author

N. J. A. Sloane, Feb 23 2006

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Crossrefs

The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. - N. J. A. Sloane, Feb 04 2020

Programs

Mathematica

a[n_]:=2 Sum[(n-i) (n-j) Boole[CoprimeQ[i,j]], {i, 1, n-1}, {j, 1, n-1}] / 2 + n^2 - n; Array[a, 40] (* Vincenzo Librandi, Feb 05 2020 *)

Python

from sympy import totient
def A115005(n): return (n-1)*(2*n-1) + sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # Chai Wah Wu, Aug 15 2021

Formula

a(n) = (n-1)*(2n-1) + Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 15 2021

Extensions

Offset corrected by Max Alekseyev, Apr 10 2019