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A161203

n-th square plus n-th squarefree number.

2, 6, 12, 21, 31, 43, 59, 75, 94, 114, 136, 161, 188, 217, 247, 279, 315, 353, 391, 431, 474, 518, 564, 613, 663, 715, 770, 826, 884, 946, 1008, 1075, 1142, 1211, 1282, 1354, 1428, 1505, 1583

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..39.

FORMULA

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

EXAMPLE

a(1) = 1^2 + 1.

a(2) = 2^2 + 2.

a(3) = 3^2 + 3.

a(4) = 4^2 + 5.

MATHEMATICA

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 *)

PROG

(Python)

from math import isqrt

from sympy import mobius

def A161203(n):

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

m, k = n, f(n)

while m != k:

m, k = k, f(k)

return m+n**2 # Chai Wah Wu, Aug 12 2024

CROSSREFS

Cf. A000290, A005117.