A005064 -id:A005064 - cp's OEIS frontend

A005067 Sum of cubes of odd primes dividing n.

0, 0, 27, 0, 125, 27, 343, 0, 27, 125, 1331, 27, 2197, 343, 152, 0, 4913, 27, 6859, 125, 370, 1331, 12167, 27, 125, 2197, 27, 343, 24389, 152, 29791, 0, 1358, 4913, 468, 27, 50653, 6859, 2224, 125, 68921, 370, 79507, 1331, 152, 12167, 103823, 27, 343, 125, 4940, 2197, 148877, 27, 1456, 343, 6886, 24389, 205379, 152

Offset: 1

N. J. A. Sloane

nonn

Harvey P. Dale (terms 1 .. 1000) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000265, A000578, A005064, A005066, A005068, A005069, A020639, A028234, A065091.

Join[{0},Table[Total[Select[Transpose[FactorInteger[n]][[1]],OddQ]^3],{n,2,50}]] (* Harvey P. Dale, Jun 09 2016 *)
Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, OddQ@ #] &] &, 60] (* Michael De Vlieger, Jul 11 2017 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005067 n) (cond ((= 1 n) 0) ((even? n) (A005067 (/ n 2))) (else (+ (A000578 (A020639 n)) (A005067 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 0 if p = 2, p^3 otherwise.
G.f.: Sum_{k>=2} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 06 2017
From Antti Karttunen, Jul 10 2017: (Start)
a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n)^3 + a(A028234(n)).
a(n) = A005064(A000265(n)).
(End)

More terms from Antti Karttunen, Jul 10 2017

A351196 Sum of the 8th powers of the primes dividing n.

Original entry on oeis.org

0, 256, 6561, 256, 390625, 6817, 5764801, 256, 6561, 390881, 214358881, 6817, 815730721, 5765057, 397186, 256, 6975757441, 6817, 16983563041, 390881, 5771362, 214359137, 78310985281, 6817, 390625, 815730977, 6561, 5765057, 500246412961, 397442, 852891037441, 256

Offset: 1

Wesley Ivan Hurt, Feb 04 2022

nonn

Inverse Möbius transform of n^8 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Cf. A010051.

Array[DivisorSum[#, #^8 &, PrimeQ] &, 50]
f[p_, e_] := p^8; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

from sympy import primefactors
def A351196(n): return sum(p**8 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(n) = Sum_{p|n, p prime} p^8.
G.f.: Sum_{k>=1} prime(k)^8 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022
Additive with a(p^e) = p^8. - Amiram Eldar, Jun 20 2022
a(n) = Sum_{d|n} d^8 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

A005080 Sum of cubes of primes = 1 mod 4 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 125, 0, 0, 0, 0, 125, 0, 0, 2197, 0, 125, 0, 4913, 0, 0, 125, 0, 0, 0, 0, 125, 2197, 0, 0, 24389, 125, 0, 0, 0, 4913, 125, 0, 50653, 0, 2197, 125, 68921, 0, 0, 0, 125, 0, 0, 0, 0, 125, 4913, 2197, 148877, 0, 125, 0, 0, 24389, 0, 125, 226981, 0, 0, 0, 2322, 0, 0, 4913, 0, 125, 0, 0, 389017, 50653, 125

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A005064, A005078, A005079, A005081, A005084, A059841.

Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, Mod[#, 4] == 1] &] &, 75] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 1, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 1, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005080 n) (if (= 1 n) 0 (+ (if (= 1 (modulo (A020639 n) 4)) (A000578 (A020639 n)) 0) (A005080 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p^3 if p = 1 (mod 4), 0 otherwise.

a(n) = A005064(n) - A005084(n) - 8*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 2017

A005084 Sum of cubes of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 27, 0, 0, 27, 343, 0, 27, 0, 1331, 27, 0, 343, 27, 0, 0, 27, 6859, 0, 370, 1331, 12167, 27, 0, 0, 27, 343, 0, 27, 29791, 0, 1358, 0, 343, 27, 0, 6859, 27, 0, 0, 370, 79507, 1331, 27, 12167, 103823, 27, 343, 0, 27, 0, 0, 27, 1331, 343, 6886, 0, 205379, 27, 0, 29791, 370, 0, 0, 1358, 300763, 0, 12194, 343, 357911

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A005064, A005080, A005082, A005083, A005085, A059841.

Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 71] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005084 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000578 (A020639 n)) 0) (A005084 (A028234 n))))) ;;  Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p^3 if p = 3 (mod 4), 0 otherwise.

a(n) = A005064(n) - A005080(n) - 8*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 2017

A005076 Sum of cubes of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 8, 0, 8, 125, 8, 0, 8, 0, 133, 1331, 8, 0, 8, 125, 8, 4913, 8, 0, 133, 0, 1339, 12167, 8, 125, 8, 0, 8, 24389, 133, 0, 8, 1331, 4921, 125, 8, 0, 8, 0, 133, 68921, 8, 0, 1339, 125, 12175, 103823, 8, 0, 133, 4913, 8, 148877, 8, 1456, 8, 0, 24397, 205379, 133, 0, 8, 0, 8, 125, 1339, 0, 4921, 12167, 133, 357911

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A005064, A005072, A005074, A005075, A005077, A008472, A079978.

Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 71] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 2, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005076 n) (if (= 1 n) 0 (+ (A000578 (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005076 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^3 if p = 2 (mod 3), 0 otherwise.

a(n) = A005064(n) - A005072(n) - 27*A079978(n). - Antti Karttunen, Jul 10 2017

More terms from Antti Karttunen, Jul 10 2017

A347157 Sum of cubes of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 0, 8, 0, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 160, 0, 8, 27, 8, 125, 35, 0, 8, 27, 133, 0, 35, 0, 8, 152, 8, 0, 35, 0, 133, 27, 8, 0, 35, 125, 351, 27, 8, 0, 160, 0, 8, 370, 8, 125, 35, 0, 8, 27, 476, 0, 35, 0, 8, 152

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001158, A005064, A005067, A333806, A333808, A339354, A347156, A347158.

Table[DivisorSum[n, #^3 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[Prime[k]^3 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^3 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A067170 Numbers n such that sum of the cubes of the distinct prime factors of n equals the sum of the cubes of the digits of n.

Original entry on oeis.org

2, 3, 5, 7, 250, 735, 2500, 25000, 250000, 1858560, 2500000, 18585600, 25000000, 91990080, 185856000, 242121642, 250000000, 919900800, 1081088775, 1390120992, 1768635648, 1858560000, 2500000000, 5435938431, 7245987840, 9199008000, 9475854336, 17996666688, 18585600000, 24214634829, 25000000000

Offset: 1

Joseph L. Pe, Feb 18 2002

If 10*m is a term (e.g. m = 25, 185856, 9199008), then 10^k * m is a term for all k >= 1. Therefore this sequence is infinite. - Amiram Eldar, Sep 28 2019

The sum of cubes of digits of a k-digit number is at most 729*k. Therefore any term with at most k digits is p-smooth where p is the largest prime < (729*k)^(1/3). - David A. Corneth, Sep 28 2019

			The prime factors of 735 are 3,5,7, the sum of whose cubes = 495 = sum of the cubes of the digits of 735; so 735 is a term of the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11790 (first 260 terms from Giovanni Resta, terms < 10^34)

Cf. A005064, A006753, A055012, A067184.

f[n_] := Module[{a, l, t, r}, a = FactorInteger[n]; l = Length[a]; t = Table[a[[i]][[1]], {i, 1, l}]; r = Sum[(t[[i]])^3, {i, 1, l}]]; g[n_] := Module[{b, m, s}, b = IntegerDigits[n]; m = Length[b]; s = Sum[(b[[i]])^3, {i, 1, m}]]; Select[Range[2, 10^6], f[ # ] == g[ # ] &]

sd(n) = my(d=digits(n)); sum(k=1, #d, d[k]^3); \\ A055012
sp(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ A005064
isok(n) = sp(n) == sd(n); \\ Michel Marcus, Sep 28 2019

a(10)-a(14) from Amiram Eldar, Sep 28 2019
a(15)-a(18) from Michel Marcus, Sep 28 2019
a(20)-a(29) from David A. Corneth, Sep 28 2019
Missing a(19) from Giovanni Resta, Sep 28 2019

A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 9, 2, 1, 0, 16, 27, 4, 5, 2, 0, 32, 81, 8, 25, 5, 1, 0, 64, 243, 16, 125, 13, 7, 1, 0, 128, 729, 32, 625, 35, 49, 2, 1, 0, 256, 2187, 64, 3125, 97, 343, 4, 3, 2, 0, 512, 6561, 128, 15625, 275, 2401, 8, 9, 7, 1, 0, 1024, 19683, 256, 78125, 793, 16807, 16, 27, 29, 11, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  0,  0,   0,    0,    0,     0,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  3,   9,   27,   81,   243,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  5,  25,  125,  625,  3125,  ...
  2,  5,  13,   35,   97,   275,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences related to sums of divisors

Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.

Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.

Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[Prime[j]^k x^Prime[j]/(1 - x^Prime[j]), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={vecsum([p^k | p<-factor(n)[,1]])}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).

A199583 a(n) is the smallest number such that the sum of the n-th powers of its distinct prime divisors is divisible by n.

Original entry on oeis.org

2, 2, 3, 2, 5, 70, 7, 2, 3, 33, 11, 1155, 13, 78, 26, 2, 17, 2156564410, 19, 6006, 26, 114, 23, 2156564410, 5, 33, 3, 1365, 29, 110, 31, 2, 62, 15, 201, 2156564410, 37, 30, 14, 961380175077106319535, 41, 1385670, 43, 2805, 26, 266, 47, 961380175077106319535

Offset: 1

Michel Lagneau, Nov 08 2011

nonn

a(n) > 1 and a(n) = n if n prime. All terms are squarefree.

			a(6) = 70 = 2*5*7; 2^6 + 5^6 + 7^6 = 133338 = 22223*6.
a(18)= 2*5*7*11*13*17*19*23*29 = 2156564410 because:
p^18 == 10, 9 (mod 18) for p = 2,3 respectively, and p^18 == 1 (mod 18) for p prime > 3. The minimum sum divisible by 18 is s = 2^18 + Sum_{k=3..10} prime(k)^18 whose residues sum to 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 18. Hence a(18) = 2156564410.

Cf. A005063, A005064, A005065, A199167.

with(numtheory): T:=array(1..50):for n from 1 to 50 do:q:=0:for k from 2 to 7000 while(q=0)do:x:=factorset(k):s:=sum(x[j]^n ,j=1..nops(x)) :if irem(s,n)=0 then printf ( "%d %d \n",n,k):q:=1:else fi:od:if q=0 then for i from 1 to n do: T[i]:=irem(ithprime(i)^n,n):od:W:=convert(T,set):n1:=nops(W):n2:=W[n1]:n3:=W[n1-1]:
s:=0:p:=1:for a from 1 to n  while(s<>n) do: if T[a]= 1 or T[a]=n2 or (T[a] = n3 and n2+n3

A279290 Sum of cubes of nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 65, 1, 217, 1, 577, 730, 1001, 1, 2009, 1, 2745, 3376, 4673, 1, 6778, 1, 9065, 9262, 10649, 1, 16345, 15626, 17577, 20413, 24761, 1, 31592, 1, 37441, 35938, 39305, 42876, 55226, 1, 54873, 59320, 73577, 1, 86310, 1, 95897, 95230, 97337, 1, 131033, 117650, 141626, 132652, 158249, 1, 183925, 166376

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

			a(4) = 65 because 4 has 2 nonprime divisors {1,4} and 1^3 + 4^3 = 65.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of divisors.

Cf. A001158, A005064, A023890, A189120.

Table[DivisorSum[n, #1^3 & ,  !PrimeQ[#1] & ], {n, 55}]
Table[DivisorSigma[3, n] - DivisorSum[n, #1^3 & , PrimeQ[#1] & ], {n, 55}]
Table[Total[Select[Divisors[n],!PrimeQ[#]&]^3],{n,60}] (* Harvey P. Dale, Aug 02 2024 *)

a(n) = {my(f = factor(n)); sigma(f, 3) - sum(i=1, #f~, f[i, 1]^3);} \\ Amiram Eldar, Jan 11 2025

a(n) = A001158(n) - A005064(n).
a(n) = 1 when n = 1 or n is prime.
a(p^k) = (p^(3*k+3) - 1)/(p^3 - 1) - p^3 when p is prime.

cp's OEIS Frontend

A005067 Sum of cubes of odd primes dividing n.

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A351196 Sum of the 8th powers of the primes dividing n.

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A005080 Sum of cubes of primes = 1 mod 4 dividing n.

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A005084 Sum of cubes of primes = 3 mod 4 dividing n.

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A005076 Sum of cubes of primes = 2 mod 3 dividing n.

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A347157 Sum of cubes of distinct prime divisors of n that are < sqrt(n).

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A067170 Numbers n such that sum of the cubes of the distinct prime factors of n equals the sum of the cubes of the digits of n.

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