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 1-3 of 3 results.

A130030 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * prime(d).

Original entry on oeis.org

2, 1, 1, 2, 3, 6, 5, 7, 9, 14, 11, 15, 17, 20, 21, 22, 27, 20, 31, 23, 33, 38, 39, 20, 45, 48, 43, 35, 53, 6, 67, 47, 65, 64, 63, 25, 85, 78, 73, 34, 99, 20, 107, 63, 45, 94, 119, 35, 113, 56, 99, 73, 137, 54, 103, 54, 117, 134, 161, -1, 163, 136, 73, 96, 113, 24, 199, 107, 159, 12, 213
Offset: 1

Views

Author

Gary W. Adamson, May 02 2007

Keywords

Comments

Old name: A129691 * A000040.
A130029 = A054523 * A000040.

Examples

			a(4) = 2 = dot product of row 4 of A129691: (-1, -1, 0, 1) and the first four primes: (2, 3, 5, 7) = (-2, -3, 0, 7) = 2.

Crossrefs

Cf. A000010, A000040, A001221, A007444, A007947, A023900, A054523, A076479, A129691, A130029, A333177.

Formula

A129691 as an infinite lower triangular matrix * A000040, the primes.
From Ilya Gutkovskiy, Mar 22 2020: (Start)
a(n) = Sum_{d|n} A023900(n/d) * prime(d).
Sum_{k=1..n} a(gcd(n,k)) = prime(n). (End)

Extensions

New name and more terms from Ilya Gutkovskiy, Mar 22 2020

A333558 a(n) = Sum_{d|n} phi(d) * prime(d).

Original entry on oeis.org

2, 5, 12, 19, 46, 41, 104, 95, 150, 165, 312, 203, 494, 365, 432, 519, 946, 545, 1208, 747, 990, 1105, 1828, 991, 1986, 1709, 2004, 1663, 3054, 1481, 3812, 2615, 3062, 3173, 3724, 2519, 5654, 4145, 4512, 3591, 7162, 3449, 8024, 4979, 5298, 6209, 9708, 4983, 9638, 6685
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 26 2020

Keywords

Crossrefs

Cf. A000010, A000040, A007445, A057660, A061150, A130029.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] Prime[d], {d, Divisors[n]}], {n, 1, 50}]
Table[Sum[Prime[n/GCD[n, k]], {k, 1, n}], {n, 1, 50}]
  • PARI
    a(n) = sumdiv(n, d, prime(d)*eulerphi(d)); \\ Michel Marcus, Mar 27 2020

Formula

G.f.: Sum_{k>=1} phi(k) * prime(k) * x^k / (1 - x^k).
a(n) = Sum_{k=1..n} prime(n/gcd(n,k)).
a(n) = Sum_{k=1..n} prime(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021

A357210 a(n) = Sum_{k=1..n} prime(k/gcd(n,k)).

Original entry on oeis.org

2, 4, 7, 11, 19, 22, 43, 46, 66, 68, 131, 90, 199, 158, 187, 223, 383, 242, 503, 320, 441, 478, 793, 436, 824, 716, 879, 734, 1373, 658, 1595, 1118, 1313, 1358, 1579, 1103, 2429, 1776, 1957, 1556, 3089, 1532, 3449, 2192, 2347, 2784, 4229, 2144, 4134, 2882, 3687, 3258, 5591
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 19 2022

Keywords

Comments

Inverse Moebius transform of A127413.

Crossrefs

Cf. A000040, A057661, A127413, A130029, A333558.

Programs

  • Mathematica
    Table[Sum[Prime[k/GCD[n, k]], {k, 1, n}], {n, 1, 53}]
  • PARI
    a(n) = sum(k=1, n, prime(k/gcd(n,k))); \\ Michel Marcus, Sep 20 2022

Formula

a(n) = Sum_{d|n} A127413(d).
Showing 1-3 of 3 results.

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

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