A011755 - OEIS

%I #40 May 08 2024 03:50:34

%S 1,3,9,17,37,49,91,123,177,217,327,375,531,615,735,863,1135,1243,1585,

%T 1745,1997,2217,2723,2915,3415,3727,4213,4549,5361,5601,6531,7043,

%U 7703,8247,9087,9519,10851,11535,12471,13111,14751,15255,17061,17941,19021,20033

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

%C 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

%C 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

%H Seiichi Manyama, <a href="/A011755/b011755.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2001 from Indranil Ghosh)

%F Asymptotically: a(n) ~ C*n^3 with C=0.20264.... - _Benoit Cloitre_, Jan 14 2002

%F Asymptotically: a(n) ~ (2/Pi^2)*n^3. - _Vladeta Jovovic_, Jan 02 2003

%F a(n) = Sum_{k=1..n} phi(k^2). - _Vaclav Kotesovec_, May 08 2024

%t Accumulate[Table[k*EulerPhi[k], {k, 1, 50}]] (* _Vaclav Kotesovec_, Sep 10 2018 *)

%o (PARI) a(n) = sum(k=1, n, k*eulerphi(k)); \\ _Michel Marcus_, Feb 13 2017

%o (Python)

%o from sympy import totient

%o def A011755(n): return sum(k*totient(k) for k in range(1,n+1)) # _Chai Wah Wu_, Feb 21 2023

%Y Partial sums of A002618.

%Y Cf. A000010, A002088, A110663, A319087.

%K nonn

%O 1,2

%A _N. J. A. Sloane_