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.

A005065 Sum of 4th powers of primes dividing n.

Original entry on oeis.org

0, 16, 81, 16, 625, 97, 2401, 16, 81, 641, 14641, 97, 28561, 2417, 706, 16, 83521, 97, 130321, 641, 2482, 14657, 279841, 97, 625, 28577, 81, 2417, 707281, 722, 923521, 16, 14722, 83537, 3026, 97, 1874161, 130337, 28642, 641, 2825761, 2498, 3418801, 14657, 706, 279857, 4879681, 97, 2401, 641, 83602, 28577, 7890481, 97
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Primes are taken without multiplicity, e.g., 12 = 2*2*3, and a(12) = 2^4+3^4 = 97. - Harvey P. Dale, Jul 16 2014
Inverse Möbius transform of n^4 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Links

Crossrefs

Column k=4 of A322080.
Cf. A000583, A005068, A005073, A005077, A005081, A005085, A008472, A059841, A079978.
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), this sequence (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

Programs

  • Maple
    A005065 := proc(n)
        add(d^4, d= numtheory[factorset](n)) ;
end proc;
seq(A005065(n),n=1..40) ; # R. J. Mathar, Nov 08 2011
  • Mathematica
    Join[{0},Table[Total[Transpose[FactorInteger[n]][[1]]^4],{n,2,40}]] (* Harvey P. Dale, Jul 16 2014 *)
Array[DivisorSum[#, #^4 &, PrimeQ] &, 54] (* Michael De Vlieger, Jul 11 2017 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^4); \\ Michel Marcus, Jul 11 2017
  • Python
    from sympy import primefactors
def a(n): return sum(p**4 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017
  • Scheme
    (define (A005065 n) (if (= 1 n) 0 (+ (A000583 (A020639 n)) (A005065 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Formula

Additive with a(p^e) = p^4.
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005068(n) + 16*A059841(n).
a(n) = A005081(n) + A005085(n) + 16*A059841(n).
a(n) = A005073(n) + A005077(n) + 81*A079978(n).
(End)
G.f.: Sum_{k>=1} prime(k)^4*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2018
a(n) = Sum_{p|n, p prime} p^4. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^4 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Extensions

More terms from Antti Karttunen, Jul 10 2017

A005074 Sum of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 0, 2, 0, 7, 11, 2, 0, 2, 5, 2, 17, 2, 0, 7, 0, 13, 23, 2, 5, 2, 0, 2, 29, 7, 0, 2, 11, 19, 5, 2, 0, 2, 0, 7, 41, 2, 0, 13, 5, 25, 47, 2, 0, 7, 17, 2, 53, 2, 16, 2, 0, 31, 59, 7, 0, 2, 0, 2, 5, 13, 0, 19, 23, 7, 71, 2, 0, 2, 5, 2, 11, 2, 0, 7, 0, 43, 83, 2, 22, 2, 29, 13, 89, 7, 0, 25, 0, 49, 5
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A005070, A005075, A005076, A005077, A005090, A008472, A079978.

Programs

  • Mathematica
    Array[DivisorSum[#, # &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 95] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p, 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)); \\ Michel Marcus, Jul 11 2017
  • Scheme
    (define (A005074 n) (if (= 1 n) 0 (+ (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0) (A005074 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Formula

Additive with a(p^e) = p if p = 2 (mod 3), 0 otherwise.
a(n) = A008472(n) - A005070(n) - 3*A079978(n). - Antti Karttunen, Jul 10 2017

Extensions

More terms from Antti Karttunen, Jul 10 2017

A005075 Sum of squares of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 4, 0, 4, 25, 4, 0, 4, 0, 29, 121, 4, 0, 4, 25, 4, 289, 4, 0, 29, 0, 125, 529, 4, 25, 4, 0, 4, 841, 29, 0, 4, 121, 293, 25, 4, 0, 4, 0, 29, 1681, 4, 0, 125, 25, 533, 2209, 4, 0, 29, 289, 4, 2809, 4, 146, 4, 0, 845, 3481, 29, 0, 4, 0, 4, 25, 125, 0, 293, 529, 29, 5041, 4, 0, 4, 25, 4, 121, 4, 0, 29, 0, 1685, 6889, 4, 314
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000290, A005063, A005071, A005074, A005076, A005077, A008472, A079978.

Programs

  • Mathematica
    Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 85] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p^2, 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^2)); \\ Michel Marcus, Jul 11 2017
  • Scheme
    (define (A005075 n) (if (= 1 n) 0 (+ (A000290 (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005075 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Formula

Additive with a(p^e) = p^2 if p = 2 (mod 3), 0 otherwise.
a(n) = A005063(n) - A005071(n) - 9*A079978(n). - Antti Karttunen, Jul 10 2017

Extensions

More terms from Antti Karttunen, Jul 10 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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