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

1,4

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

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.

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

Example

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.

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 *)

Prog

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, !isprime(d)*d)) /* Michael Somos, Jun 08 2005 */
(Haskell)
a023890 n = sum $ zipWith (*) divs $ map ((1 -) . a010051) divs
            where divs = a027750_row n
-- Reinhard Zumkeller, Apr 12 2014
(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

Crossrefs

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

Sequence in context: A340089 A089027 A354432 * A319684 A102778 A135544

Adjacent sequences: A023887 A023888 A023889 * A023891 A023892 A023893

Keyword

nonn,nice,easy

Author

Olivier Gérard

Status

approved