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

Offset: 1

Author

Jonathan Vos Post, Jan 20 2011

Keywords

• nonn
• easy

Examples

			a(1) = 1^2 + 1.
a(2) = 2^2 + 2.
a(3) = 3^2 + 3.
a(4) = 4^2 + 5.
		

Crossrefs

Cf. A000290, A005117.

Programs

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 *)

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

Formula

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

Extensions

Corrected (75 inserted) by Harvey P. Dale, May 06 2012