A213519 -id:A213519 - cp's OEIS frontend

A005064 Sum of cubes of primes dividing n.

0, 8, 27, 8, 125, 35, 343, 8, 27, 133, 1331, 35, 2197, 351, 152, 8, 4913, 35, 6859, 133, 370, 1339, 12167, 35, 125, 2205, 27, 351, 24389, 160, 29791, 8, 1358, 4921, 468, 35, 50653, 6867, 2224, 133, 68921, 378, 79507, 1339, 152, 12175, 103823, 35, 343, 133, 4940, 2205, 148877, 35, 1456, 351, 6886, 24397, 205379, 160

Offset: 1

N. J. A. Sloane

nonn

The set of these terms is A213519. - Bernard Schott, Feb 11 2022

Inverse Möbius transform of n^3 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

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

Cf. A000578, A005067, A005072, A005076, A005080, A005084, A059841, A079978.
Cf. A010051, A213519.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), this sequence (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Array[DivisorSum[#, #^3 &, PrimeQ] &, 60] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := p^3; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

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

from sympy import primefactors
def a(n): return sum(p**3 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005064 n) (if (= 1 n) 0 (+ (A000578 (A020639 n)) (A005064 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^3.
G.f.: Sum_{k>=1} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005067(n) + 8*A059841(n).
a(n) = A005080(n) + A005084(n) + 8*A059841(n).
a(n) = A005072(n) + A005076(n) + 27*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-3)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^3. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^3 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Antti Karttunen, Jul 10 2017

A121571 Largest number that is not the sum of n-th powers of distinct primes.

Original entry on oeis.org

6, 17163, 1866000

Offset: 1

T. D. Noe, Aug 08 2006

As stated by Sierpinski, H. E. Richert proved a(1) = 6. Dressler et al. prove a(2) = 17163.
Fuller & Nichols prove T. D. Noe's conjecture that a(3) = 1866000. They also prove that 483370 positive numbers cannot be written as the sum of cubes of distinct primes. - Robert Nichols, Sep 08 2017
Noe conjectures that a(4) = 340250525752 and that 31332338304 positive numbers cannot be written as the sum of fourth powers of distinct primes. - Charles R Greathouse IV, Nov 04 2017

			a(1) = 6 because only the numbers 1, 4 and 6 are not the sum of distinct primes.

W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964, p. 143-144.

R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.
Robert E. Dressler, Louis Pigno and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18, (2015), #15.10.5.
H. E. Richert, Über Zerfällungen in ungleiche Primzahlen, Math. Z. 52 no. 1 (1948), 342-343.

Cf. A231407 (numbers that are not the sum of distinct primes).
Cf. A121518 (numbers that are not the sum of squares of distinct primes).
Cf. A213519 (numbers that are the sum of cubes of distinct primes).
Cf. A001661 (integers instead of primes).

a(1) = A231407(3), a(2) = A121518(2438). - Jonathan Sondow, Nov 26 2013

A351326 a(n) is the least number k such that k and all larger numbers can be expressed as the sum of n-th powers of distinct primes.

Original entry on oeis.org

7, 17164, 1866001

Offset: 1

Ilya Gutkovskiy, Mar 24 2022

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.

Cf. A048261, A121571, A213519, A334360.

a(n) = A121571(n) + 1.

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A005064 Sum of cubes of primes dividing n.

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A121571 Largest number that is not the sum of n-th powers of distinct primes.

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A351326 a(n) is the least number k such that k and all larger numbers can be expressed as the sum of n-th powers of distinct primes.

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