A076664 a(n) = Sum_{k=1..n} antisigma(k), where antisigma(i) = sum of the nondivisors of i that are between 1 and i.

0, 0, 2, 5, 14, 23, 43, 64, 96, 133, 187, 237, 314, 395, 491, 596, 731, 863, 1033, 1201, 1400, 1617, 1869, 2109, 2403, 2712, 3050, 3400, 3805, 4198, 4662, 5127, 5640, 6181, 6763, 7338, 8003, 8684, 9408, 10138, 10957, 11764, 12666, 13572, 14529, 15538

Offset: 1

Author

Joseph L. Pe, Oct 24 2002

Keywords

• nonn
• easy

Comments

Sum of all proper nondivisors of all positive integers <= n. - Omar E. Pol, Feb 11 2014

Examples

			a(5) = antisigma(1) + ... + antisigma(5) = 0 + 0 + 2 + 3 + 9 = 14.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A024816, A000292, A024916.

Programs

Mathematica

l = {}; s = 0; Do[s = s + (n (n + 1) / 2) - DivisorSigma[1, n]; l = Append[l, s], {n, 1, 100}]; l
Accumulate[Table[Total[Complement[Range[n],Divisors[n]]],{n,50}]] (* Harvey P. Dale, May 19 2014 *)

PARI

a(n) = sum(k=1, n, k*(k+1)/2-sigma(k)); \\ Michel Marcus, Sep 18 2017

Python

from math import isqrt
def A076664(n): return n*(n+1)*(n+2)//3+(s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 22 2023

Formula

a(n) = A000292(n) - A024916(n), n >= 1. Omar E. Pol, Feb 11 2014

a(n) = Sum_{k=1..n} Sum_{i=1..k-1} (n-k-i+1) mod (n-i+1). - Wesley Ivan Hurt, Sep 13 2017

G.f.: x/(1 - x)^4 - (1/(1 - x))*Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017