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n-th square plus n-th squarefree number.
2

%I #28 Aug 13 2024 02:25:58

%S 2,6,12,21,31,43,59,75,94,114,136,161,188,217,247,279,315,353,391,431,

%T 474,518,564,613,663,715,770,826,884,946,1008,1075,1142,1211,1282,

%U 1354,1428,1505,1583

%N n-th square plus n-th squarefree number.

%F a(n) = n^2 + A005117(n) = A000290(n) + A005117(n).

%e a(1) = 1^2 + 1.

%e a(2) = 2^2 + 2.

%e a(3) = 3^2 + 3.

%e a(4) = 4^2 + 5.

%t Module[{nn=40,sqs,sfree},sqs=Range[nn]^2;sfree=Take[Select[Range[3nn], SquareFreeQ],nn];Total/@Thread[{sqs,sfree}]] (* _Harvey P. Dale_, May 06 2012 *)

%o (Python)

%o from math import isqrt

%o from sympy import mobius

%o def A161203(n):

%o def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return m+n**2 # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A000290, A005117.

%K nonn,easy

%O 1,1

%A _Jonathan Vos Post_, Jan 20 2011

%E Corrected (75 inserted) by _Harvey P. Dale_, May 06 2012