A035321 -id:A035321 - cp's OEIS frontend

A023888 Sum of prime power divisors of n (1 included).

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 12, 10, 14, 10, 9, 31, 18, 15, 20, 12, 11, 14, 24, 18, 31, 16, 40, 14, 30, 11, 32, 63, 15, 20, 13, 19, 38, 22, 17, 20, 42, 13, 44, 18, 18, 26, 48, 34, 57, 33, 21, 20, 54, 42, 17, 22, 23, 32, 60, 15, 62, 34, 20, 127, 19, 17, 68, 24, 27

Offset: 1

Olivier Gérard

nonn

Sum of n-th row of triangle A210208. [Reinhard Zumkeller, Mar 18 2012]

			For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1+2+3+4 = 10. From

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

Cf. A008475, A159077.

a023888 = sum . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

f:= n -> 1 + add((t[1]^(t[2]+1)-t[1])/(t[1]-1),t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Jan 04 2017

Array[ Plus @@ (Select[ Divisors[ # ], (Length[ FactorInteger[ # ] ]<=1)& ])&, 70 ]

for(n=1,100, s=1; fordiv(n,d, if((ispower(d,,&z)&&isprime(z)) || isprime(d),s+=d)); print1(s,", "))

a(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  1 + sum(k = 1, fsz, f[k,1]*(f[k,1]^f[k,2] - 1)\(f[k,1]-1));
};
vector(100, n, a(n))  \\ Gheorghe Coserea, Jan 04 2017

a(n) = A000203(n) - A035321(n) = A023889(n) + 1.
a(1) = 1, a(p) = p+1, a(pq) = p+q+1, a(pq...z) = (p+q+...+z) + 1, a(p^k) = (p^(k+1)-1) / (p-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*k*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

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

A023891 Sum of composite divisors of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 6, 0, 12, 9, 10, 0, 22, 0, 14, 15, 28, 0, 33, 0, 34, 21, 22, 0, 54, 25, 26, 36, 46, 0, 61, 0, 60, 33, 34, 35, 85, 0, 38, 39, 82, 0, 83, 0, 70, 69, 46, 0, 118, 49, 85, 51, 82, 0, 114, 55, 110, 57, 58, 0, 157, 0, 62, 93, 124, 65, 127, 0, 106, 69, 129, 0

Offset: 1

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000203, A035322, A035321, A060278, A023890.

Array[ Plus @@ (Select[ Divisors[ # ], (!PrimeQ[ # ] && #>1)& ])&, 75 ]
a[n_] := DivisorSigma[1, n] - Plus @@ FactorInteger[n][[;; , 1]] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = sumdiv(n, d, d*!isprime(d)) - 1; \\ Michel Marcus, Jun 12 2019

a(n) = A023890(n) - 1. - Sean A. Irvine, Jun 11 2019

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

A178637 a(n) = sum of divisors d of n such that d is not equal to p^k where p = prime, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 19, 1, 15, 16, 1, 1, 25, 1, 31, 22, 23, 1, 43, 1, 27, 1, 43, 1, 62, 1, 1, 34, 35, 36, 73, 1, 39, 40, 71, 1, 84, 1, 67, 61, 47, 1, 91, 1, 61, 52, 79, 1, 79, 56, 99, 58, 59, 1, 154, 1, 63, 85, 1, 66, 128, 1, 103, 70, 130, 1, 169, 1, 75, 91, 115, 78, 150, 1, 151, 1, 83, 1, 208, 86, 87, 88, 155, 1, 215, 92, 139, 94, 95, 96, 187, 1, 113, 133, 181

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

			For n = 12, set of such divisors is {1, 6, 12}; a(12) = 1+6+12 = 19.

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

One more than A035321.

Cf. A000203, A001221 (omega), A023889, A035321.

Array[Plus @@ (Select[Divisors[#], (Length[FactorInteger[#]] > 1) &]) &, 100] + 1 (* Robert P. P. McKone, Jan 28 2021 *)

A178637(n) = sumdiv(n,d,(omega(d)!=1)*(d)); \\ Antti Karttunen, Aug 06 2018

a(n) = A000203(n) - A023889(n) = A035321(n) + 1.
a(1) = 1, a(p) = 1, a(pq) = pq+1, a(pq...z) = [(p+1)*(q+1)*…*(z+1)] - (p+q+...+z), a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
a(n) = Sum_{d|n} d * (1 - [omega(n) = 1]), where omega is the number of distinct prime factors (A001221) and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

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A023888 Sum of prime power divisors of n (1 included).

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A060278 Sum of composite divisors of n less than n.

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A023891 Sum of composite divisors of n.

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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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A178637 a(n) = sum of divisors d of n such that d is not equal to p^k where p = prime, k >=1.

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