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A023890 Sum of nonprime divisors of n. 13
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 (list; graph; refs; listen; history; text; internal format)
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)

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #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

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 *)

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 isprime(i)==False:

............s+=i

....return s # Indranil Ghosh, Jan 30 2017

CROSSREFS

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

Sequence in context: A318676 A265293 A089027 * A319684 A102778 A135544

Adjacent sequences:  A023887 A023888 A023889 * A023891 A023892 A023893

KEYWORD

nonn,nice,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified April 23 18:15 EDT 2019. Contains 322387 sequences. (Running on oeis4.)