A115005 - OEIS

%I #25 Aug 16 2021 13:45:35

%S 0,3,14,43,100,209,374,641,1020,1553,2246,3197,4372,5911,7778,10037,

%T 12728,16043,19862,24467,29728,35777,42626,50625,59520,69675,80966,

%U 93627,107568,123345,140458,159673,180664,203651,228590,255857,285116,317363,352058

%N a(n) = (A114043(n) - 1)/2.

%H Seiichi Manyama, <a href="/A115005/b115005.txt">Table of n, a(n) for n = 1..1000</a>

%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

%F a(n) = (n-1)*(2n-1) + Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - _Chai Wah Wu_, Aug 15 2021

%t 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 *)

%o (Python)

%o from sympy import totient

%o 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

%Y 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

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Feb 23 2006

%E Offset corrected by _Max Alekseyev_, Apr 10 2019