A363421 a(n) = Sum_{k=0..n}(n^[not(k | n)] - n^[k | n]), where '[ ]' denotes the Iverson bracket.

1, 0, -1, 0, -3, 8, -5, 24, 7, 32, 27, 80, 11, 120, 91, 112, 105, 224, 119, 288, 171, 280, 315, 440, 207, 480, 475, 520, 459, 728, 435, 840, 651, 832, 891, 952, 665, 1224, 1147, 1216, 975, 1520, 1107, 1680, 1419, 1496, 1755, 2024, 1363, 2112, 1911, 2200, 2091

Offset: 0

Author

Peter Luschny, Jun 27 2023

Keywords

• sign

Links

• Peter Luschny, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000005, A113704, A363734, A363735.

Programs

Mathematica

A363421[n_]:=If[n==0,1,n^2-2(n-1)DivisorSigma[0,n]-1];Array[A363421,100,0] (* Paolo Xausa, Aug 06 2023 *)

Python

from sympy import divisor_count
def A363421(n): return n**2-2*(n-1)*divisor_count(n)-1 if n else 1 # Chai Wah Wu, Jun 28 2023

SageMath

print([sum(n^(not k.divides(n)) - n^k.divides(n) for k in srange(n+1)) for n in srange(53)])

Formula

a(n) = n^2 - 2*(n - 1)*tau(n) - 1 for n >= 1, where tau = A000005.