A023889 -id:A023889 - 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

A333753 Sum of prime power divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 5, 0, 2, 3, 6, 0, 5, 0, 6, 3, 2, 0, 9, 5, 2, 3, 6, 0, 10, 0, 6, 3, 2, 5, 9, 0, 2, 3, 11, 0, 5, 0, 6, 8, 2, 0, 9, 7, 7, 3, 6, 0, 5, 5, 13, 3, 2, 0, 14, 0, 2, 10, 14, 5, 5, 0, 6, 3, 14, 0, 17, 0, 2, 8, 6, 7, 5, 0, 19, 12, 2, 0, 16, 5, 2, 3, 14, 0, 19

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A023889, A066839, A069289, A069293, A097974, A246655, A333750, A333751, A333752.

f:= proc(n) local F,i,j,t;
  F:= ifactors(n)[2];
  t:= 0;
  for i from 1 to nops(F) do
    j:= min(F[i,2],ilog[F[i,1]^2](n));
    t:= t + (F[i,1]^j-1)*F[i,1]/(F[i,1]-1)
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Feb 15 2023

Table[DivisorSum[n, # &, # <= Sqrt[n] && PrimePowerQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && isprimepower(d), d)); \\ Michel Marcus, Apr 03 2020

G.f.: Sum_{p prime, k>=1} p^k * x^(p^(2*k)) / (1 - x^(p^k)).

A284117 Sum of proper prime power divisors of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 28, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 60, 0, 0, 0, 13, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 28, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 124, 0, 0, 0, 4, 0, 0, 0, 21, 0, 0, 25, 4, 0, 0, 0, 28, 117, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 60, 0, 49, 9, 29

Offset: 1

Ilya Gutkovskiy, Mar 20 2017

			a(8) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 2 are proper prime powers {4, 8} therefore 4 + 8 = 12.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Cf. A001414, A008472, A023888, A023889, A046660, A246547.

f:= n -> add(t[1]*(t[1]^t[2]-t[1])/(t[1]-1), t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Mar 31 2017

nmax = 100; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#1] && PrimeOmega[#1] > 1 &]], {n, 100}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p - 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

concat([0, 0, 0], Vec(sum(k=1, 100, (isprimepower(k) && bigomega(k)>1) * k * x^k/(1 - x^k)) + O(x^101))) \\ Indranil Ghosh, Mar 21 2017

a(n) = sumdiv(n, d, d*(isprimepower(d) && !isprime(d))); \\ Michel Marcus, Apr 01 2017

G.f.: Sum_{p prime, k>=2} p^k*x^(p^k)/(1 - x^(p^k)).
a(n) = Sum_{d|n, d = p^k, p prime, k >= 2} d.
a(n) = 0 if n is a squarefree (A005117).
Additive with a(p^e) = (p^(e+1)-1)/(p-1) - p - 1. - Amiram Eldar, Jul 24 2024

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

A281782 Numbers n such that sum of prime power divisors of n > sum of prime power divisors of m for all m < n.

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 27, 32, 64, 121, 125, 128, 243, 256, 512, 729, 1024, 2048, 4096, 6561, 8192, 15625, 16384, 32761, 32768, 59049, 65536, 117649, 130321, 131072, 177147, 262144, 524287, 524288, 1048576, 1594323, 1953125, 2097152, 4194304, 8388608

Offset: 1

Ilya Gutkovskiy, Apr 14 2017

nonn

Numbers n such that A023889(n) > A023889(m) for all m < n.

Numbers n such that Sum_{p^k|n, p prime, k>=1} p^k > Sum_{p^k|m, p prime, k>=1} p^k for all m < n.

Amiram Eldar, Table of n, a(n) for n = 1..65
Index entries for sequences related to sums of divisors

Cf. A002093, A002182, A023889, A034090, A174572, A246655, A280013.

mx = 0; t = {}; Do[u = DivisorSum[n, # &, PrimePowerQ[#] &]; If[u > mx, mx = u; AppendTo[t, n]], {n, 8500000}]; t

A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 8, 9, 0, 0, 4, 0, 0, 0, 16, 0, 9, 0, 4, 0, 0, 0, 8, 25, 0, 27, 4, 0, 0, 0, 32, 0, 0, 0, 13, 0, 0, 0, 8, 0, 0, 0, 4, 9, 0, 0, 16, 49, 25, 0, 4, 0, 27, 0, 8, 0, 0, 0, 4, 0, 0, 9, 64, 0, 0, 0, 4, 0, 0, 0, 17, 0, 0, 25, 4, 0, 0, 0, 16, 81, 0, 0, 4, 0, 0, 0, 8, 0, 9, 0, 4, 0, 0, 0, 32, 0, 49, 9, 29, 0, 0, 0, 8, 0, 0, 0, 31

Offset: 1

Ilya Gutkovskiy, Aug 02 2017

Sum of unitary, proper prime power divisors of n.

			a(360) = a(2^3*3^2*5) = 2^3 + 3^2 = 17.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A005117, A008475, A222416, A023888, A023889, A034448, A063956, A077610, A092261, A246547, A284117.

Table[DivisorSum[n, # &, CoprimeQ[#, n/#] && PrimePowerQ[#] && !PrimeQ[#] &], {n, 108}]
f[p_, e_] := If[e == 1, 0, p^e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

A286875(n) = { my(f=factor(n)); for (i=1, #f~, if(f[i, 2] < 2, f[i, 1] = 0)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Oct 07 2017

from sympy import primefactors, isprime, gcd, divisors
def a(n): return sum(d for d in divisors(n) if gcd(d, n//d)==1 and len(primefactors(d))==1 and not isprime(d))
print([a(n) for n in range(1, 109)]) # Indranil Ghosh, Aug 02 2017

a(n) = Sum_{d|n, d = p^k, p prime, k >= 2, gcd(d, n/d) = 1} d.
a(A246547(k)) = A246547(k).
a(A005117(k)) = 0.
Additive with a(p^e) = p^e if e >= 2, and 0 otherwise. - Amiram Eldar, Jul 24 2024

A284233 Sum of odd prime power divisors of n (not including 1).

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Ilya Gutkovskiy, Mar 23 2017

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.

Eric Weisstein's World of Mathematics, Prime Power.
Index entries for sequences related to sums of divisors.

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).
a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.
a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.
a(2^k*p) = p for p is a prime > 2.
a(2^k) = 0.
Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024

A286972 Numbers k such that the average of the prime power divisors (not including 1) of k is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 42, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 119, 121, 123, 127, 129, 131, 132, 133, 135, 137, 139

Offset: 1

Ilya Gutkovskiy, May 17 2017

nonn

Numbers k such that A001222(k)|A023889(k).

			12 is in the sequence because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are prime powers {2, 3, 4} and (2 + 3 + 4)/3 = 3 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A001222, A003601, A023886, A023889, A078174, A246655, A285510.

fQ[n_] := IntegerQ@Mean@Select[Divisors@n, PrimePowerQ]; Select[Range@140, fQ]

isok(m) = my(vd = select(isprimepower, divisors(m))); #vd && !(vecsum(vd) % #vd); \\ Michel Marcus, Apr 28 2020

A351395 Sum of the divisors of n that are either squarefree, prime powers, or both.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 16, 14, 24, 24, 31, 18, 21, 20, 22, 32, 36, 24, 24, 31, 42, 40, 28, 30, 72, 32, 63, 48, 54, 48, 25, 38, 60, 56, 30, 42, 96, 44, 40, 33, 72, 48, 40, 57, 43, 72, 46, 54, 48, 72, 36, 80, 90, 60, 76, 62, 96, 41, 127, 84, 144, 68, 58, 96, 144, 72

Offset: 1

Wesley Ivan Hurt, Feb 09 2022

nonn

			a(36) = 25; 36 has 4 squarefree divisors 1,2,3,6 (where the primes 2 and 3 are both squarefree and 1st powers of primes) and 2 (additional) divisors that are powers of primes, 2^2 and 3^2. The sum of the divisors is then 1+2+3+4+6+9 = 25.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A008683, A246655.

Sums of divisors: A048250 (squarefree), A023889 (prime powers), A008472 (prime).

Array[DivisorSum[#, #*Sign[MoebiusMu[#]^2 + Boole[PrimeNu[#] == 1]] &] &, 71] (* Michael De Vlieger, Feb 10 2022 *)

a(n) = sumdiv(n, d, if (issquarefree(d) || isprimepower(d), d)); \\ Michel Marcus, Feb 10 2022

a(n) = Sum_{d|n} d * sign(mu(d)^2 + [omega(d) = 1]).

A281906 Expansion of Sum_{p prime, i>=1} p^i*x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).

Original entry on oeis.org

0, 2, 5, 13, 23, 41, 69, 119, 185, 283, 425, 625, 903, 1285, 1799, 2517, 3450, 4699, 6340, 8490, 11264, 14870, 19485, 25390, 32897, 42395, 54372, 69408, 88210, 111612, 140717, 176738, 221135, 275776, 342790, 424743, 524765, 646420, 794109, 972967, 1189105, 1449577, 1763097, 2139394, 2590349, 3129633, 3773546, 4540645

Offset: 1

Ilya Gutkovskiy, Feb 01 2017

nonn

Total sum of prime power parts (1 excluded) in all partitions of n.

Convolution of the sequences A000041 and A023889.

			a(5) = 23 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 5 + 4 + 3 + 2 + 3 + 2 + 2 + 2 = 23.

Index entries for related partition-counting sequences

Cf. A000041, A023889, A066186, A073118, A246655.

nmax = 48; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[i]] i x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{p prime, i>=1} p^i*x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).

cp's OEIS Frontend

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

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A333753 Sum of prime power divisors of n that are <= sqrt(n).

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A284117 Sum of proper prime power divisors of n.

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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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A281782 Numbers n such that sum of prime power divisors of n > sum of prime power divisors of m for all m < n.

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A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

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A284233 Sum of odd prime power divisors of n (not including 1).

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