A023890 Sum of the nonprime divisors of n.

1, 1, 1, 5, 1, 7, 1, 13, 10, 11, 1, 23, 1, 15, 16, 29, 1, 34, 1, 35, 22, 23, 1, 55, 26, 27, 37, 47, 1, 62, 1, 61, 34, 35, 36, 86, 1, 39, 40, 83, 1, 84, 1, 71, 70, 47, 1, 119, 50, 86, 52, 83, 1, 115, 56, 111, 58, 59, 1, 158, 1, 63, 94, 125, 66, 128, 1, 107, 70, 130, 1, 190, 1, 75

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• nice
• easy

Comments

Obviously a(n) < sigma(n) for all n > 1, where sigma(n) is the sum of divisors function (A000203). It thus follows that a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Mar 16 2013

Examples

			a(8) = 13 because the divisors of 8 are 1, 2, 4, 8, and without the 2 they add up to 13.
a(9) = 10 because the divisors of 9 are 1, 3, 9, and without the 3 they add up to 10.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe)
• Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.

Crossrefs

Cf. A000203, A001222, A010051, A023891, A027750, A051731, A037282.

Programs

Haskell

a023890 n = sum $ zipWith (*) divs $ map ((1 -) . a010051) divs
            where divs = a027750_row n
-- Reinhard Zumkeller, Apr 12 2014

Mathematica

Array[ Plus @@ (Select[ Divisors[ # ], (!PrimeQ[ # ])& ])&, 75 ]
Table[DivisorSum[n, # &, Not[PrimeQ[#]] &], {n, 75}] (* Alonso del Arte, Mar 16 2013 *)
Table[CoefficientList[Series[Log[Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, 100}]], {x, 0, 100}], x][[n + 1]] n, {n, 1, 100}] (* Benedict W. J. Irwin, Jul 05 2016 *)
a[n_] := DivisorSigma[1, n] - Plus @@ FactorInteger[n][[;; , 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

PARI

a(n)=if(n<1, 0, sumdiv(n,d, !isprime(d)*d)) /* Michael Somos, Jun 08 2005 */

Python

from sympy import isprime
def A023890(n):
    s=0
    for i in range(1,n+1):
        if n%i==0 and not isprime(i):
            s+=i
    return s # Indranil Ghosh, Jan 30 2017

Formula

Equals A051731 * A037282. - Gary W. Adamson, Nov 06 2007

a(n) = A023891(n) + 1 (sum of composite divisors of n + 1). [Alonso del Arte, Oct 01 2008]

a(n) = A000203(n) - A008472(n). - R. J. Mathar, Aug 14 2011

a(n) = Sum (a027750(n,k)*(1-A010051(a027750(n,k))): k=1..A000005(n)). - Reinhard Zumkeller, Apr 12 2014

L.g.f.: log(Product_{ k>0 } (1-x^prime(k))/(1-x^k)) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016

a(n) = Sum_{d|n} d * (1 - [Omega(d) = 1]), where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021