cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A005064 Sum of cubes of primes dividing n.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ Michel Marcus, Jul 11 2017
  • Python
    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
  • Scheme
    (define (A005064 n) (if (= 1 n) 0 (+ (A000578 (A020639 n)) (A005064 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Formula

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

Extensions

More terms from Antti Karttunen, Jul 10 2017

A005071 Sum of squares of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 169, 49, 0, 0, 0, 0, 361, 0, 49, 0, 0, 0, 0, 169, 0, 49, 0, 0, 961, 0, 0, 0, 49, 0, 1369, 361, 169, 0, 0, 49, 1849, 0, 0, 0, 0, 0, 49, 0, 0, 169, 0, 0, 0, 49, 361, 0, 0, 0, 3721, 961, 49, 0, 169, 0, 4489, 0, 0, 49, 0, 0, 5329, 1369, 0, 361, 49, 169, 6241, 0, 0, 0, 0, 49, 0, 1849, 0, 0, 0, 0, 218
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000290, A005070, A005072, A005073.

Programs

  • Mathematica
    Module[{sp=Select[Prime[Range[100]],Mod[#,3]==1&]},Table[Total[ Select[ sp, Divisible[ n,#]&]^2],{n,70}]] (* Harvey P. Dale, Dec 19 2014 *)
f[p_, e_] := If[Mod[p, 3] == 1, p^2, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)
  • Scheme
    (define (A005071 n) (if (= 1 n) 0 (+ (A000290 (if (= 1 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005071 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

Formula

Additive with a(p^e) = p^2 if p = 1 (mod 3), 0 otherwise.

Extensions

More terms from Antti Karttunen, Jul 09 2017

A005073 Sum of 4th powers of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2401, 0, 0, 0, 0, 0, 28561, 2401, 0, 0, 0, 0, 130321, 0, 2401, 0, 0, 0, 0, 28561, 0, 2401, 0, 0, 923521, 0, 0, 0, 2401, 0, 1874161, 130321, 28561, 0, 0, 2401, 3418801, 0, 0, 0, 0, 0, 2401, 0, 0, 28561, 0, 0, 0, 2401, 130321, 0, 0, 0, 13845841, 923521, 2401, 0, 28561, 0, 20151121, 0, 0, 2401, 0, 0, 28398241, 1874161
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000583, A005070, A005071, A005072.

Programs

  • Mathematica
    f[p_, e_] := If[Mod[p, 3] == 1, p^4, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 1, p^4)); \\ Michel Marcus, Jul 10 2017
  • Scheme
    (define (A005073 n) (if (= 1 n) 0 (+ (A000583 (if (= 1 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005073 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

Formula

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

Extensions

More terms from Antti Karttunen, Jul 09 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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

More terms from Antti Karttunen, Jul 10 2017
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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