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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A069913 a(n) = Sum_{d|n} (d-1)*tau(n/d).

Original entry on oeis.org

0, 1, 2, 5, 4, 11, 6, 16, 12, 19, 10, 37, 12, 27, 26, 42, 16, 54, 18, 59, 36, 43, 22, 100, 32, 51, 48, 81, 28, 113, 30, 99, 56, 67, 54, 162, 36, 75, 66, 152, 40, 153, 42, 125, 108, 91, 46, 240, 60, 134, 86, 147, 52, 202, 82, 204, 96, 115, 58, 331, 60, 123, 144, 219, 96
Offset: 1

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Author

Vladeta Jovovic, May 04 2002

Keywords

Crossrefs

Cf. A000005, A006579, A007425, A007429.

Programs

  • Mathematica
    Table[DivisorSum[n,(#-1)DivisorSigma[0,n/#]&],{n,100}] (* Giorgos Kalogeropoulos, Aug 19 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (d-1)*numdiv(n/d)) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = A007429(n) - A007425(n).
G.f.: Sum_{k>=1} tau(k) * x^(2*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Aug 19 2021

A106473 Rows of A003989 expressed as base 10 numbers.

Original entry on oeis.org

1, 11, 121, 1111, 12321, 111111, 1214121, 11311311, 121252121, 1111111111, 12341614321, 111111111111, 1212127212121, 11315311351311, 121412181214121, 1111111111111111, 12321612921612321, 111111111111111111
Offset: 0

Views

Author

Paul Barry, May 03 2005

Keywords

Comments

The all-1 entries occur at positions indexed by the n-th primes-2 (A040976). The 1's then sum to give n-th prime-1 (A006093). The sum of the digits of a(n) is given by A006579(n+1).

Formula

a(n) = sum{k=0..n, GCD(n-k+1, k+1)*10^k}.

A326824 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * d * a(d).

Original entry on oeis.org

1, 1, 2, 4, 4, 12, 6, 24, 18, 32, 10, 124, 12, 60, 72, 240, 16, 336, 18, 440, 132, 140, 22, 2088, 100, 192, 378, 1044, 28, 2096, 30, 4320, 300, 320, 312, 9636, 36, 396, 408, 10384, 40, 5040, 42, 3500, 3000, 572, 46, 61584, 294, 3920, 672, 5544, 52, 23148, 680
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 23 2020

Keywords

Crossrefs

Cf. A000010, A001710, A006579, A006874, A018804, A165552.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[n/d] d a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
a[1] = 1; a[n_] := Sum[GCD[n, k] a[GCD[n, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 55}]

Formula

a(1) = 1; a(n) = Sum_{k=1..n-1} gcd(n, k) * a(gcd(n, k)).
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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