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

1,3

Comments

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

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A024816, A000292, A024916.

Sequence in context: A265248 A131661 A321287 * A220477 A049939 A296302

Adjacent sequences: A076661 A076662 A076663 * A076665 A076666 A076667

Keyword

nonn,easy

Author

Joseph L. Pe, Oct 24 2002

Status

approved