A024816 Antisigma(n): Sum of the numbers less than n that do not divide n.

0, 0, 2, 3, 9, 9, 20, 21, 32, 37, 54, 50, 77, 81, 96, 105, 135, 132, 170, 168, 199, 217, 252, 240, 294, 309, 338, 350, 405, 393, 464, 465, 513, 541, 582, 575, 665, 681, 724, 730, 819, 807, 902, 906, 957, 1009, 1080, 1052, 1168, 1182, 1254, 1280, 1377, 1365

Offset: 1

Author

Paul Jobling (paul.jobling(AT)whitecross.com)

Keywords

• easy
• nonn
• nice

Comments

a(n) is the sum of proper non-divisors of n, the row sum in triangle A173541. - Omar E. Pol, May 25 2010

a(n) is divisible by A000203(n) iff n is in A076617. - Bernard Schott, Apr 12 2022

Examples

			a(12)=50 as 5+7+8+9+10+11 = 50 (1,2,3,4,6 not included as they divide 12).
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203 (sigma), A000217, A004125, A023896, A024916, A066760, A076617, A153485, A173539, A173540, A173541, A244048, A352810, A352811.

Cf. A342344 (for a symmetric representation).

Programs

Haskell

a024816 = sum . a173541_row  -- Reinhard Zumkeller, Feb 19 2014

Magma

[n*(n+1) div 2- SumOfDivisors(n): n in [1..60]]; // Vincenzo Librandi, Dec 29 2015

Maple

A024816 := proc(n)
    n*(n+1)/2-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Aug 03 2013

Mathematica

Table[n(n + 1)/2 - DivisorSigma[1, n], {n, 55}] (* Robert G. Wilson v *)
Table[Total[Complement[Range[n],Divisors[n]]],{n,60}] (* Harvey P. Dale, Sep 23 2012 *)
With[{nn=60},#[[1]]-#[[2]]&/@Thread[{Accumulate[Range[nn]],DivisorSigma[ 1,Range[nn]]}]] (* Harvey P. Dale, Nov 22 2014 *)

PARI

a(n)=n*(n+1)/2-sigma(n) \\ Charles R Greathouse IV, Mar 19 2012

Python

from sympy import divisor_sigma
def A024816(n): return (n*(n+1)>>1)-divisor_sigma(n) # Chai Wah Wu, Apr 28 2023

SageMath

def A024816(n): return sum(k for k in (0..n-1) if not k.divides(n))
print([A024816(n) for n in srange(1, 55)])  # Peter Luschny, Nov 14 2023

Formula

a(n) = n*(n+1)/2 - sigma(n) = A000217(n) - A000203(n).

a(n) = A024916(n-1) - A153485(n), n > 1. - Omar E. Pol, Jun 24 2014

From Wesley Ivan Hurt, Jul 16 2014, Dec 28 2015: (Start)

a(n) = Sum_{i=1..n} i * ( ceiling(n/i) - floor(n/i) ).

a(n) = Sum_{k=1..n} (n mod k) + (-n mod k). (End)

G.f.: x/(1 - x)^3 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017

From Omar E. Pol, Mar 21 2021: (Start)

a(n) = A244048(n) + A004125(n).

a(n) = A153485(n-1) + A004125(n), n >= 2. (End)

a(p) = (p-2)*(p+1)/2 for p prime. - Bernard Schott, Apr 12 2022