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 1-6 of 6 results.

A369911 a(n) = Sum_{p|n, p prime} p * sopf(n/p).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 4, 9, 20, 0, 16, 0, 28, 30, 4, 0, 21, 0, 24, 42, 44, 0, 16, 25, 52, 9, 32, 0, 62, 0, 4, 66, 68, 70, 25, 0, 76, 78, 24, 0, 82, 0, 48, 39, 92, 0, 16, 49, 45, 102, 56, 0, 21, 110, 32, 114, 116, 0, 66, 0, 124, 51, 4, 130, 122, 0, 72, 138, 118, 0, 25
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Comments

Dirichlet convolution of A061397(n) and A008472(n). - Wesley Ivan Hurt, Jul 10 2025

Links

Crossrefs

Cf. A008472 (sopf), A061397, A369744.

Programs

  • Mathematica
    a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[DivisorSum[n, #*a[n/#] &, PrimeQ[#] &], {n, 100}]
  • PARI
    A008472(n) = vecsum(factor(n)[, 1]);
A369911(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i,1]*A008472(n/f[i, 1]))); \\ Antti Karttunen, Jan 23 2025

Formula

From Wesley Ivan Hurt, Jul 10 2025: (Start)
a(n) = Sum_{d|n} A061397(d) * A008472(n/d).
a(p^k) = p^2 * (1-floor(1/k)) for p prime and k>=1. (End)

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

A369907 a(n) = n * Sum_{p|n, p prime} omega(n/p) / p.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 4, 3, 7, 0, 16, 0, 9, 8, 8, 0, 21, 0, 24, 10, 13, 0, 32, 5, 15, 9, 32, 0, 62, 0, 16, 14, 19, 12, 60, 0, 21, 16, 48, 0, 82, 0, 48, 39, 25, 0, 64, 7, 45, 20, 56, 0, 63, 16, 64, 22, 31, 0, 154, 0, 33, 51, 32, 18, 122, 0, 72, 26, 118, 0, 120, 0, 39
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Links

Crossrefs

Cf. A001221 (omega), A345354, A345355, A369744.
Cf. also A369894, A369909, A369779.

Programs

  • Mathematica
    Table[n*DivisorSum[n, PrimeNu[n/#]/# &, PrimeQ[#] &], {n, 100}]
  • PARI
    A369907(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (omega(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 23 2025

Formula

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

A369912 a(n) = Sum_{p|n, p prime} p^sopf(n/p).

Original entry on oeis.org

0, 1, 1, 4, 1, 17, 1, 4, 27, 57, 1, 41, 1, 177, 368, 4, 1, 251, 1, 153, 2530, 2169, 1, 41, 3125, 8361, 27, 561, 1, 5568, 1, 4, 178478, 131361, 94932, 275, 1, 524649, 1596520, 153, 1, 37514, 1, 8313, 6686, 8389137, 1, 41, 823543, 78157, 129145076, 32937, 1, 251
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Crossrefs

Cf. A008472 (sopf), A369744, A369911, A369913.

Programs

  • Mathematica
    a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[DivisorSum[n, #^a[n/#] &, PrimeQ[#] &], {n, 80}]

Formula

a(p^k) = p^(p-p*floor(1/k)) for p prime and k>=1. - Wesley Ivan Hurt, Jul 09 2025

A369913 a(n) = Sum_{p|n, p prime} n^sopf(n/p).

Original entry on oeis.org

0, 1, 1, 16, 1, 252, 1, 64, 729, 100100, 1, 248976, 1, 105413700, 762750, 256, 1, 1895400, 1, 1280000400, 1801097802, 584318301411812, 1, 7963200, 9765625, 2481152873203737252, 19683, 10578455954192, 1, 677994300000, 1, 1024, 50542106513762754
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Crossrefs

Cf. A008472 (sopf), A369744, A369911, A369912.

Programs

  • Mathematica
    a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[DivisorSum[n, n^a[n/#] &, PrimeQ[#] &], {n, 40}]

Formula

a(p^k) = p^(p*(k-floor(1/k))) for p prime and k>=1. - Wesley Ivan Hurt, Jul 09 2025

A369914 a(n) = n * Sum_{p|n, p prime} sopf(n/p) / p.

Original entry on oeis.org

0, 0, 0, 4, 0, 13, 0, 8, 9, 29, 0, 38, 0, 53, 34, 16, 0, 57, 0, 78, 58, 125, 0, 76, 25, 173, 27, 134, 0, 220, 0, 32, 130, 293, 74, 150, 0, 365, 178, 156, 0, 366, 0, 294, 147, 533, 0, 152, 49, 195, 298, 398, 0, 171, 146, 268, 370, 845, 0, 500, 0, 965, 237, 64, 194
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Comments

Dirichlet convolution of c(n) and n * sopf(n), where c = A010051. - Wesley Ivan Hurt, Jul 10 2025

Crossrefs

Cf. A008472 (sopf), A010051, A369744, A369911, A369912, A369913.

Programs

  • Mathematica
    a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[n*DivisorSum[n, a[n/#]/# &, PrimeQ[#] &], {n, 80}]

Formula

a(p^k) = p^k * (1 - floor(1/k)) for p prime and k>=1. - Wesley Ivan Hurt, Jul 09 2025
a(n) = n * Sum_{d|n} c(d) * sopf(n/d) / d, where c = A010051. - Wesley Ivan Hurt, Jul 10 2025
Showing 1-6 of 6 results.

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