A023891 -id:A023891 - cp's OEIS frontend

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

Olivier Gérard

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

			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.

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.

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

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

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 *)

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

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

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

A060278 Sum of composite divisors of n less than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 12, 0, 15, 0, 14, 0, 0, 0, 30, 0, 0, 9, 18, 0, 31, 0, 28, 0, 0, 0, 49, 0, 0, 0, 42, 0, 41, 0, 26, 24, 0, 0, 70, 0, 35, 0, 30, 0, 60, 0, 54, 0, 0, 0, 97, 0, 0, 30, 60, 0, 61, 0, 38, 0, 59, 0, 117, 0, 0, 40, 42, 0, 71, 0, 98, 36, 0, 0, 127, 0, 0, 0

Offset: 1

Jack Brennen, Mar 28 2001

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A010051, A027751, A033942, A035322, A035321, A037143, A023891.

a060278 1 = 0
a060278 n = sum $ filter ((== 0) . a010051) $ tail $ a027751_row n
-- Reinhard Zumkeller, Apr 05 2013

for n from 1 to 300 do s := 0: for j from 2 to n-1 do if isprime(j) then else if n mod j = 0 then s := s+j fi; fi: od: printf(`%d,`,s) od:

Join[{0},Table[Total[Select[Most[Rest[Divisors[n]]],!PrimeQ[#]&]],{n,2,90}]] (* Harvey P. Dale, Oct 25 2011 *)
a[n_] := DivisorSigma[1, n] - Plus @@ FactorInteger[n][[;; , 1]] - If[PrimeQ[n], 0, n] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = sumdiv(n, d, if ((d1) && !isprime(d), d)); \\ Michel Marcus, Jan 13 2020

From Reinhard Zumkeller, Apr 05 2013: (Start)
a(n) = Sum_{k=2..A000005(n)-1} A010051(A027751(n,k));
a(A037143(n)) = 0;
a(A033942(n)) > 0. (End)

More terms from James Sellers and Matthew Conroy, Mar 29 2001

A035321 Sum of composite divisors of n that are not primes nor prime powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 10, 0, 18, 0, 14, 15, 0, 0, 24, 0, 30, 21, 22, 0, 42, 0, 26, 0, 42, 0, 61, 0, 0, 33, 34, 35, 72, 0, 38, 39, 70, 0, 83, 0, 66, 60, 46, 0, 90, 0, 60, 51, 78, 0, 78, 55, 98, 57, 58, 0, 153, 0, 62, 84, 0, 65, 127, 0, 102, 69, 129, 0, 168, 0, 74, 90, 114, 77

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A035322, A023891, A060278.

One less than A178637.

pp := array(1..100); for i from 1 to 100 do pp[i] := 0; od: for i from 1 to 25 do for j from 1 to 6 do t1 := ithprime(i)^j; if t1<100 then pp[t1] := 1; fi; od: od: pp[1] := 1; A035321 := proc(n) local i,d,t1,t2; t1 := 0; for d from 1 to n do if n mod d = 0 and pp[d] = 0 then t1 := t1+d; fi; od; t1; end;

Array[ Plus @@ (Select[ Divisors[ # ], (Length[ FactorInteger[ # ] ]>1)& ])&, 80 ]

A035321(n) = sumdiv(n,d,(omega(d)>1)*(d)); \\ Antti Karttunen, Aug 06 2018

a(n) = A178637(n) - 1. - Antti Karttunen, Aug 06 2018

Description corrected by Jack Brennen, Mar 28 2001

A035322 Sum of composite divisors of n that are less than n and are not primes nor prime powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 10, 0, 0, 0, 18, 0, 0, 0, 14, 0, 31, 0, 0, 0, 0, 0, 36, 0, 0, 0, 30, 0, 41, 0, 22, 15, 0, 0, 42, 0, 10, 0, 26, 0, 24, 0, 42, 0, 0, 0, 93, 0, 0, 21, 0, 0, 61, 0, 34, 0, 59, 0, 96, 0, 0, 15, 38, 0, 71, 0, 70, 0, 0, 0, 123, 0, 0, 0, 66, 0

Offset: 1

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A035321, A023891, A060278.

Table[ Plus @@ Select[ Divisors[ n ], (Length[ FactorInteger[ # ] ]>1 && #Harvey P. Dale, Jan 09 2019 *)

Description corrected by Jack Brennen, Mar 28 2001

A239668 Sum of the composite divisors of n^2.

Original entry on oeis.org

0, 4, 9, 28, 25, 85, 49, 124, 117, 209, 121, 397, 169, 389, 394, 508, 289, 841, 361, 953, 730, 917, 529, 1645, 775, 1265, 1089, 1757, 841, 2810, 961, 2044, 1714, 2129, 1754, 3745, 1369, 2645, 2362, 3929, 1681, 5174, 1849, 4109, 3742, 3845, 2209, 6637, 2793, 5459, 3970

Offset: 1

Janet Lee, Mar 23 2014

nonn

			For n=2, the sum of the composite factors of n^2 is equal to 4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A023891.

A008472 := n -> add(d, d = select(isprime, numtheory[divisors](n))):
f:=n->numtheory[sigma](n^2)-A008472(n)-1; [seq(f(n),n=1..100)]; # N. J. A. Sloane, Mar 31 2014

a[n_] := DivisorSum[n^2, If[# == 1 || PrimeQ[#], 0, #]& ]; Array[a, 60] (* Jean-François Alcover, Dec 18 2015 *)
Table[Total[Select[Divisors[n^2],CompositeQ]],{n,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 06 2017 *)

a(n) = sumdiv(n^2, d, d*(!isprime(d) && (d != 1))); \\ Michel Marcus, Mar 31 2014

a(n) = sigma(n^2) - sopf(n^2) - 1.
a(n) = A000203(n^2) - A008472(n^2) - 1. - Wesley Ivan Hurt, Mar 30 2014
a(n) = A023891(n^2). - Michel Marcus, Mar 31 2014
a(n) = n^2 if n is prime. - Zak Seidov, Mar 31 2014

Formula corrected by Wesley Ivan Hurt, Mar 30 2014

More terms from N. J. A. Sloane, Mar 31 2014

cp's OEIS Frontend

A023890 Sum of the nonprime divisors of n.

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A060278 Sum of composite divisors of n less than n.

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A035321 Sum of composite divisors of n that are not primes nor prime powers.

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A035322 Sum of composite divisors of n that are less than n and are not primes nor prime powers.

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A239668 Sum of the composite divisors of n^2.

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