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.

A369748 a(n) = Sum_{p|n, p prime} prime(n/p).

Original entry on oeis.org

0, 2, 2, 3, 2, 8, 2, 7, 5, 14, 2, 20, 2, 20, 16, 19, 2, 36, 2, 36, 22, 34, 2, 56, 11, 44, 23, 50, 2, 89, 2, 53, 36, 62, 28, 98, 2, 70, 46, 90, 2, 129, 2, 86, 70, 86, 2, 142, 17, 126, 64, 108, 2, 164, 42, 126, 72, 112, 2, 221, 2, 130, 96, 131, 52, 229, 2, 146, 88, 221
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 30 2024

Keywords

Links

Crossrefs

Cf. A000040, A351368.

Programs

  • Mathematica
    Table[DivisorSum[n, Prime[n/#] &, PrimeQ[#] &], {n, 100}]
  • PARI
    A369748(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, prime(n/f[i, 1]))); \\ Antti Karttunen, Jan 22 2025

A369749 a(n) = Sum_{p|n, p prime} p * prime(n/p).

Original entry on oeis.org

0, 4, 6, 6, 10, 19, 14, 14, 15, 37, 22, 47, 26, 55, 58, 38, 34, 85, 38, 93, 86, 95, 46, 131, 55, 121, 69, 135, 58, 246, 62, 106, 148, 169, 162, 233, 74, 191, 188, 237, 82, 366, 86, 235, 256, 235, 94, 337, 119, 339, 262, 293, 106, 389, 276, 347, 296, 305, 118, 624
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 30 2024

Keywords

Crossrefs

Cf. A000040, A351368, A369748.

Programs

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

Formula

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

A351369 a(n) = Sum_{p|n, p prime} p * prime(p).

Original entry on oeis.org

0, 6, 15, 6, 55, 21, 119, 6, 15, 61, 341, 21, 533, 125, 70, 6, 1003, 21, 1273, 61, 134, 347, 1909, 21, 55, 539, 15, 125, 3161, 76, 3937, 6, 356, 1009, 174, 21, 5809, 1279, 548, 61, 7339, 140, 8213, 347, 70, 1915, 9917, 21, 119, 61, 1018, 539, 12773, 21, 396, 125, 1288, 3167
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 08 2022

Keywords

Comments

Inverse Möbius transform of n * prime(n) * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

Examples

			a(6) = 21; a(6) = Sum_{p|6} p * prime(p) = 2*3 + 3*5 = 21.

Links

Crossrefs

Cf. A000040, A010051, A033286, A061397, A351368.

Programs

  • Mathematica
    Join[{0},Table[Total[# Prime[#]&/@FactorInteger[n][[;;,1]]],{n,2,80}]] (* Harvey P. Dale, Jan 28 2024 *)

Formula

a(n) = Sum_{d|n} d * prime(d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025
a(p^k) = p * prime(p) for p prime and k>=1. - Wesley Ivan Hurt, Jul 16 2025

A369750 a(n) = Sum_{p|n, p prime} p^prime(n/p).

Original entry on oeis.org

0, 4, 9, 8, 25, 59, 49, 128, 243, 2173, 121, 10379, 169, 131415, 180272, 524288, 289, 9982931, 361, 536949037, 129156970, 2147484979, 529, 138601214939, 48828125, 2199023257749, 94143178827, 8796093845751, 841, 209369086423336, 961, 9007199254740992
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 30 2024

Keywords

Crossrefs

Cf. A000040, A351368, A369748, A369749.

Programs

  • Mathematica
    Table[DivisorSum[n, #^Prime[n/#] &, PrimeQ[#] &], {n, 40}]

Formula

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

A369866 a(n) = Sum_{p|n, p prime} n^prime(n/p).

Original entry on oeis.org

0, 4, 9, 64, 25, 7992, 49, 2097152, 59049, 100000001000, 121, 106993241210880, 169, 30491346729331198648, 8649756618750, 75557863725914323419136, 289, 74347713614042750878184448000, 361, 53687091200000000000000000001280000000, 30041942495081695978842, 412195366437884247746798137865015318817176
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 03 2024

Keywords

Crossrefs

Cf. A000040, A351368, A369748.

Programs

  • Mathematica
    Table[DivisorSum[n, n^Prime[n/#] &, PrimeQ[#] &], {n, 30}]

Formula

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

A369867 a(n) = n * Sum_{p|n, p prime} prime(n/p) / p.

Original entry on oeis.org

0, 2, 2, 6, 2, 21, 2, 28, 15, 61, 2, 106, 2, 125, 70, 152, 2, 285, 2, 318, 134, 347, 2, 596, 55, 539, 207, 630, 2, 1073, 2, 848, 356, 1009, 174, 1542, 2, 1279, 548, 1572, 2, 2213, 2, 1766, 912, 1915, 2, 2984, 119, 2715, 1018, 2654, 2, 3879, 396, 3148, 1288, 3167
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 03 2024

Keywords

Crossrefs

Cf. A000040, A351368, A369748, A369749, A369750, A369866.
Cf. A003557, A033286, A246655.

Programs

  • Mathematica
    Table[n*DivisorSum[n, Prime[n/#]/# &, PrimeQ[#] &], {n, 60}]
  • PARI
    a(n) = my(vp=primes([1, n])); n*sum(i=1, #vp, if (!(n % vp[i]), prime(n/vp[i])/vp[i])); \\ Michel Marcus, May 11 2024

Formula

From Wesley Ivan Hurt, May 10 2024: (Start)
a(p^k) = p^(k-1) * prime(p^(k-1)) for primes p and k >= 1.
a(A246655(n)) = A033286(A003557(A246655(n))). (End)
Showing 1-6 of 6 results.

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