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.

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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

A333471 a(n) = 2 * mu(n) + Sum_{d|n, d > 1} mu(n/d) * (prime(d) - prime(d-1)).

Original entry on oeis.org

2, -1, 0, 1, 2, 1, 2, 0, 2, 3, 0, 3, 2, -1, 0, 4, 4, -2, 4, -3, -2, 5, 2, 0, 4, 1, -2, 1, 0, -3, 12, -2, 4, -3, 4, -4, 4, 1, 0, 2, 4, 1, 8, -5, -2, -1, 10, 2, 0, -8, -2, 1, 0, 10, 2, 2, 0, 1, 4, -1, 0, -3, 10, 0, -4, -7, 12, 3, 6, -9, 2, 4, 6, 1, -2, -3, 2, 3, 2, -2
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 23 2020

Keywords

Comments

Moebius transform of A054541 (2 followed by prime gaps).

Crossrefs

Cf. A000040, A001223, A007444, A008683, A030013, A054541, A182986, A333450 (partial sums).

Programs

  • Mathematica
    a[n_] := 2 MoebiusMu[n] + Sum[If[d > 1, MoebiusMu[n/d] (Prime[d] - Prime[d - 1]), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 80}]

Formula

a(n) = Sum_{d|n} mu(n/d) * A054541(d).
Sum_{k=1..n} floor(n/k) * a(k) = prime(n).

A352915 Moebius transform of odd primes.

Original entry on oeis.org

3, 2, 4, 6, 10, 8, 16, 12, 22, 16, 34, 18, 40, 26, 36, 36, 58, 28, 68, 36, 56, 44, 86, 44, 88, 58, 78, 56, 110, 48, 128, 78, 98, 86, 122, 66, 160, 94, 126, 94, 178, 76, 190, 108, 124, 120, 220, 94, 210, 114, 174, 132, 248, 112, 216, 148, 196, 162, 278, 96
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 26 2022

Keywords

Crossrefs

Cf. A007444, A030013, A065091, A353078.

Programs

  • Mathematica
    Table[DivisorSum[n, MoebiusMu[n/#] Prime[# + 1] &], {n, 1, 60}]
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)* prime(d+1)); \\ Michel Marcus, Apr 27 2022

Formula

Sum_{n>=1} a(n) * x^n / (1 - x^n) = Sum_{n>=1} prime(n+1) * x^n.
a(n) = Sum_{d|n} mu(n/d)* prime(d+1).
Previous Showing 11-13 of 13 results.

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

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