A175361 - OEIS

%I #14 Jun 21 2024 17:23:22

%S 1,13,73,233,485,797,1341,2301,3321,4197,5757,8157,10237,12277,15541,

%T 19701,23793,27273,31653,38853,45405,50013,58173,68733,76957,84769,

%U 94969,108089,120569,130673,144817,164017,180397,191917,209317,234277,252673,269113,293593

%N Partial sums of A000141.

%C The 6th row of A122510.

%F a(n^2) = A055412(n).

%F G.f.: theta_3(x)^6 / (1 - x). - _Ilya Gutkovskiy_, Feb 13 2021

%t CoefficientList[Series[EllipticTheta[3,x]^6/(1-x),{x,0,35}],x] (* _Stefano Spezia_, Jun 21 2024 *)

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A175361(n):

%o     c = 1

%o     for m in range(1,n+1):

%o         f = [(p,e,(0,1,0,-1)[p&3]) for p,e in factorint(m).items()]

%o         c += (prod((p**(e+1<<1)-a)//(p**2-a) for p, e, a in f)<<2)-prod(((k:=p**2*a)**(e+1)-1)//(k-1) for p, e, a in f)<<2

%o     return c # _Chai Wah Wu_, Jun 21 2024

%Y Cf. A000141, A055412, A122510.

%K nonn

%O 0,2

%A _R. J. Mathar_, Apr 24 2010