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.

A345302 a(n) = Sum_{p|n, p prime} lcm(p,n/p).

Original entry on oeis.org

0, 2, 3, 2, 5, 12, 7, 4, 3, 20, 11, 18, 13, 28, 30, 8, 17, 24, 19, 30, 42, 44, 23, 36, 5, 52, 9, 42, 29, 90, 31, 16, 66, 68, 70, 30, 37, 76, 78, 60, 41, 126, 43, 66, 60, 92, 47, 72, 7, 60, 102, 78, 53, 72, 110, 84, 114, 116, 59, 150, 61, 124, 84, 32, 130, 198, 67, 102, 138, 210
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

Comments

If p is prime, a(p) = Sum_{p|p} lcm(p,p/p) = p.

Examples

			a(12) = Sum_{p|12} lcm(p,12/p) = lcm(2,6) + lcm(3,4) = 6 + 12 = 18.

Links

Crossrefs

Cf. A008472, A057670, A345266, A369915.

Programs

  • Mathematica
    Table[Sum[LCM[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

Formula

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

A345305 a(n) = n * Sum_{p|n, p prime} gcd(p,n/p) / p.

Original entry on oeis.org

0, 1, 1, 4, 1, 5, 1, 8, 9, 7, 1, 16, 1, 9, 8, 16, 1, 27, 1, 24, 10, 13, 1, 32, 25, 15, 27, 32, 1, 31, 1, 32, 14, 19, 12, 72, 1, 21, 16, 48, 1, 41, 1, 48, 54, 25, 1, 64, 49, 75, 20, 56, 1, 81, 16, 64, 22, 31, 1, 92, 1, 33, 72, 64, 18, 61, 1, 72, 26, 59, 1, 144, 1, 39, 100, 80, 18
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

Comments

If p is prime, a(p) = p * Sum_{p|p} gcd(p,p/p) / p = p * (1/p) = 1.

Examples

			a(18) = 18 * Sum_{p|18} gcd(p,18/p) / p = 18 * (gcd(2,9)/2 + gcd(3,6)/3) = 18 * (1/2 + 1) = 27.

Links

Crossrefs

Cf. A345266.

Programs

  • Mathematica
    Table[n*Sum[(1/k) GCD[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
  • PARI
    A345305(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (gcd(f[i,1],n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 24 2025

A369915 a(n) = Sum_{p|n, p prime} lcm(p, n/p) / p.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 2, 1, 7, 1, 7, 1, 9, 8, 4, 1, 11, 1, 9, 10, 13, 1, 14, 1, 15, 3, 11, 1, 31, 1, 8, 14, 19, 12, 13, 1, 21, 16, 18, 1, 41, 1, 15, 14, 25, 1, 28, 1, 27, 20, 17, 1, 33, 16, 22, 22, 31, 1, 47, 1, 33, 16, 16, 18, 61, 1, 21, 26, 59, 1, 26, 1, 39, 28, 23, 18
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

Links

Crossrefs

Cf. A332619, A345266, A345302.

Programs

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

Formula

a(p^k) = p^(k-2+floor(1/k)) for p prime and k>=1. - Wesley Ivan Hurt, Jul 09 2025

A345303 a(n) = Sum_{p|n, p prime} p * gcd(p,n/p).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 4, 9, 7, 11, 7, 13, 9, 8, 4, 17, 11, 19, 9, 10, 13, 23, 7, 25, 15, 9, 11, 29, 10, 31, 4, 14, 19, 12, 13, 37, 21, 16, 9, 41, 12, 43, 15, 14, 25, 47, 7, 49, 27, 20, 17, 53, 11, 16, 11, 22, 31, 59, 12, 61, 33, 16, 4, 18, 16, 67, 21, 26, 14, 71, 13, 73, 39, 28
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

Comments

If p is prime, a(p) = Sum_{p|p} p * gcd(p,p/p) = p * 1 = p.

Examples

			a(18) = Sum_{p|18} p * gcd(p,18/p) = 2*gcd(2,9) + 3*gcd(3,6) = 2*1 + 3*3 = 11.

Crossrefs

Cf. A345266.

Programs

  • Mathematica
    Table[Sum[k*GCD[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

A345304 a(n) = Sum_{p|n, p prime} p * lcm(p,n/p).

Original entry on oeis.org

0, 4, 9, 4, 25, 30, 49, 8, 9, 70, 121, 48, 169, 126, 120, 16, 289, 54, 361, 120, 210, 286, 529, 96, 25, 390, 27, 224, 841, 300, 961, 32, 462, 646, 420, 72, 1369, 798, 624, 240, 1681, 504, 1849, 528, 270, 1150, 2209, 192, 49, 150, 1020, 728, 2809, 162, 880, 448, 1254, 1798, 3481
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

Comments

If p is prime, a(p) = Sum_{p|p} p * lcm(p,p/p) = p * p = p^2.

Examples

			a(18) = Sum_{p|18} p * lcm(p,18/p) = 2*lcm(2,9) + 3*lcm(3,6) = 2*18 + 3*6 = 54.

Links

Crossrefs

Cf. A008472, A345266, A345302.

Programs

  • Mathematica
    Table[Sum[k*LCM[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

A351748 a(n) = Sum_{p|n, p prime} n^gcd(p,n/p).

Original entry on oeis.org

0, 2, 3, 16, 5, 12, 7, 64, 729, 20, 11, 156, 13, 28, 30, 256, 17, 5850, 19, 420, 42, 44, 23, 600, 9765625, 52, 19683, 812, 29, 90, 31, 1024, 66, 68, 70, 47952, 37, 76, 78, 1640, 41, 126, 43, 1980, 91170, 92, 47, 2352, 678223072849, 312500050, 102, 2756, 53, 157518, 110
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 17 2022

Keywords

Examples

			a(12) = 156; a(12) = Sum_{p|12, p prime} 12^gcd(p,12/p) = 12^gcd(2,12/2) + 12^gcd(3,12/3) = 12^2 + 12^1 = 156.

Crossrefs

Cf. A345266.
Showing 1-6 of 6 results.

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