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.

A345355 a(n) = Sum_{p|n, p prime} p^omega(n/p).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 2, 3, 7, 1, 7, 1, 9, 8, 2, 1, 11, 1, 9, 10, 13, 1, 7, 5, 15, 3, 11, 1, 38, 1, 2, 14, 19, 12, 13, 1, 21, 16, 9, 1, 62, 1, 15, 14, 25, 1, 7, 7, 27, 20, 17, 1, 11, 16, 11, 22, 31, 1, 42, 1, 33, 16, 2, 18, 134, 1, 21, 26, 78, 1, 13, 1, 39, 28, 23, 18, 182
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 15 2021

Keywords

Examples

			a(30) = Sum_{p|30} p^omega(30/p) = 2^omega(15) + 3^omega(10) + 5^omega(6) = 2^2 + 3^2 + 5^2 = 38.

Links

Crossrefs

Cf. A001221 (omega), A010051, A369744.
Cf. also A369741, A369905, A369907.

Programs

  • Mathematica
    Table[Sum[k^PrimeNu[n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^omega(n/f[k,1])); \\ Michel Marcus, Jun 16 2021

Formula

a(p) = 1 for p prime.
a(n) = Sum_{d|n} d^omega(n/d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 13 2025

A369909 a(n) = n * Sum_{p|n, p prime} Omega(n/p) / p.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 8, 3, 7, 0, 20, 0, 9, 8, 24, 0, 30, 0, 28, 10, 13, 0, 60, 5, 15, 18, 36, 0, 62, 0, 64, 14, 19, 12, 90, 0, 21, 16, 84, 0, 82, 0, 52, 48, 25, 0, 160, 7, 70, 20, 60, 0, 135, 16, 108, 22, 31, 0, 186, 0, 33, 60, 160, 18, 122, 0, 76, 26, 118, 0, 240, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Links

Crossrefs

Cf. A001222 (Omega), A369741, A369908.
Cf. also A369894.

Programs

  • Mathematica
    Table[n*DivisorSum[n, PrimeOmega[n/#]/# &, PrimeQ[#] &], {n, 100}]
  • PARI
    A369909(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (bigomega(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 22 2025

Formula

a(p^k) = (k-1)*p^(k-1), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

A369908 a(n) = Sum_{p|n, p prime} n^Omega(n/p).

Original entry on oeis.org

0, 1, 1, 4, 1, 12, 1, 64, 9, 20, 1, 288, 1, 28, 30, 4096, 1, 648, 1, 800, 42, 44, 1, 27648, 25, 52, 729, 1568, 1, 2700, 1, 1048576, 66, 68, 70, 93312, 1, 76, 78, 128000, 1, 5292, 1, 3872, 4050, 92, 1, 10616832, 49, 5000, 102, 5408, 1, 314928, 110, 351232, 114
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Crossrefs

Cf. A001222 (Omega), A369741.

Programs

  • Mathematica
    Table[DivisorSum[n, n^PrimeOmega[n/#] &, PrimeQ[#] &], {n, 100}]

Formula

a(p^k) = p^(k*(k-1)), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

A369743 a(n) = Sum_{p|n, p prime} p * Omega(n/p).

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 4, 3, 7, 0, 10, 0, 9, 8, 6, 0, 10, 0, 14, 10, 13, 0, 15, 5, 15, 6, 18, 0, 20, 0, 8, 14, 19, 12, 15, 0, 21, 16, 21, 0, 24, 0, 26, 16, 25, 0, 20, 7, 14, 20, 30, 0, 15, 16, 27, 22, 31, 0, 30, 0, 33, 20, 10, 18, 32, 0, 38, 26, 28, 0, 20, 0, 39, 16
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 30 2024

Keywords

Crossrefs

Cf. A001222 (Omega), A329347, A369741.

Programs

  • Mathematica
    Table[DivisorSum[n, #*PrimeOmega[n/#] &, PrimeQ[#] &], {n, 100}]

Formula

a(p^k) = p*(k-1), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.