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 1-4 of 4 results.

A248614 Rank of the n-th distinct value of the Bernoulli denominators in the sequence of the denominators of the Bernoulli numbers.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 12, 16, 18, 20, 22, 28, 30, 36, 40, 42, 44, 46, 48, 52, 58, 60, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 106, 108, 110, 112, 116, 120, 126, 130, 132, 136, 138, 140, 144, 148, 150, 156, 162, 164, 166, 172, 174, 176, 178, 180, 190, 192
Offset: 0

Views

Author

Paul Curtz, Oct 09 2014

Keywords

Comments

Consider sequence A027642 of the denominators of the Bernoulli numbers and the reduced sequence b(n) = 1, 2, 6, 30, 42, 66,... if duplicates are removed (which is 1, 2 followed by A090126). a(n) shows the smallest index --place of first appearance-- of b(n) in the full list A027642.
If n is of the form A002322(p*q) with p*q semiprime, then n is a term. The number 3652 is a term, but it is not of the form A002322(p*q), as Carl Pomerance noted. - Thomas Ordowski, Apr 28 2021; in place of an incorrect comment by Filip Zaludek, Sep 23 2016
For n > 0, numbers n such that A002322(A027642(n)) = n. - Thomas Ordowski, Jul 11 2018
Carl Pomerance (in answer to my question) proved that the set of these numbers has asymptotic density zero. - Thomas Ordowski, Apr 28 2021

Examples

			b(2)=6 appears first in A027642(2), so a(2)=2. b(4)=42 appears first as A027642(6)=42, so a(4)=6. b(5)=66 appears first as A027642(10), so a(5)=10.

Crossrefs

Cf. A002322, A002445, A027642, A090126, A090801, A317210.

Programs

  • Mathematica
    BB = Table[Denominator[BernoulliB[n]], {n, 2, 400, 2}]; For[t = BB; n = 1, n <= Length[t], n++, p = Position[t, t[[n]]] // Rest; t = Delete[t, p]]; reducedBB = Join[{1, 2}, t]; a[0] = 0; a[1] = 1; a[n_] := 2*Position[BB, reducedBB[[n+1]], 1, 1][[1, 1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 16 2014 *)
  • PARI
    L=List(); N=60; forprime(p=2, N*N, forprime(q=p, N*N, listput(L, lcm(p-1,q-1)) )); listsort(L, 1); for (i=1, N, print1(L[i], ", ")) \\ Filip Zaludek, Sep 23 2016

A114649 Denominators of BernoulliB ranked by frequency of occurrence.

Original entry on oeis.org

6, 30, 42, 66, 510, 138, 798, 2730, 870, 282, 330, 354, 1806, 498, 1590
Offset: 1

Views

Author

Eric W. Weisstein, Dec 21 2005

Keywords

Comments

a(n+1) mod 9 = 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6 which appears to be A010704(n+1). - Paul Curtz, Oct 28 2012

Links

Crossrefs

Cf. A002445, A090126, A090801, A100194.

Extensions

Additional 10 terms from Eric W. Weisstein link, Oct 28 2012

A248913 A248614(n+1) - A248614(n).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 4, 2, 2, 2, 6, 2, 6, 4, 2, 2, 2, 2, 4, 6, 2, 6, 4, 2, 6, 2, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 2, 2, 4, 4, 6, 4, 2, 4, 2, 2, 4, 4, 2, 6, 6, 2, 2, 6, 2, 2, 2, 2, 10, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 2, 2, 4, 2, 6, 4, 2, 4, 4, 2, 2, 4, 2
Offset: 0

Views

Author

Paul Curtz, Oct 16 2014

Keywords

Comments

From Bernoulli numbers A027642.
The first different numbers of a(n), i.e., 1, 2, 4, 6, 10, ... = A248614(1, 2, 3, 4, 5, ...) are at rank 0, 2, 4, 10, 58, ... .

Links

Crossrefs

Cf. A027642, A090126, A248614.

Programs

  • PARI
    lista() = {vbden = readvec("c:/gp/bfiles/b027642.txt"); vredu = readvec("c:/gp/bfiles/b090126.txt"); vrank = []; for (i=1, #vredu, val = vredu[i]; k = 1; while(vbden[k] != val, k++); vrank = concat(vrank, k-1);); for (i=2, #vrank, print1(vrank[i] - vrank[i-1], ", "););} \\ Michel Marcus, Nov 08 2014

Extensions

More terms from Michel Marcus, Nov 08 2014

A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

Original entry on oeis.org

1, 2, 6, 1, 30, 2, 42, 1, 30, 2, 66, 1, 2730, 2, 6, 1, 510, 2, 798, 1, 330, 2, 138, 1, 2730, 2, 6, 1, 870, 2, 14322, 1, 510, 2, 6, 1, 1919190, 2, 6, 1, 13530, 2, 1806, 1, 690, 2, 282, 1, 46410, 2, 66, 1, 1590, 2, 798, 1, 870, 2, 354, 1
Offset: 0

Views

Author

Paul Curtz, Oct 28 2014

Keywords

Comments

There exist an infinity of 1's, 2's, 6's, 30's, 42's, 66's, ... .
Respective ranks:
0, 3, 7, 11, 15, 19, ...
1, 5, 9, 13, 17, 21, ... (= A016813)
2, 14, 26, 34, 38, 62, ... (= A051222)
4, 8, 68, 76, 124, 152, ... (= A051226)
6, 114, 186, 258, 354, 402, ... (= A051228)
10, 50, 170, 370, 470, 590, ... (= A051230)
12, 24, 1308, 1884, 2004, 2364, ... (= A249134)
etc.
Hence by antidiagonals a permutation of A001477(n).
First column: A248614(n).
a(n) is an alternative sequence for the denominators of the Bernoulli numbers.
First 36 terms of the corresponding clockwise spiral:
.
330------2----138------1---2730------2
| |
| |
1 42------1-----30------2 6
| | | |
| | | |
798 2 1------2 66 1
| | | | |
| | | | |
2 30------1------6 1 870
| | |
| | |
510------1------6------2---2730 2
|
|
1------6------2----510------1--14322

Links

Crossrefs

A variant of the Clausen numbers A141056, A160014. And of A176591.
Cf. A000034, A002445, A016813, A027642, A051222, A051226, A051228, A051230, A090126, A164020, A248614, A249134.

Programs

  • Maple
    Clausen := proc(n) local S, i;
S := numtheory[divisors](n); S := map(i->i+1, S);
S := select(isprime, S); mul(i, i=S) end:
A249306 := n -> `if`(n mod 4 = 3, 1, Clausen(n)):
seq(A249306(n), n=0..59); # Peter Luschny, Nov 10 2014
  • Mathematica
    a[n_] := Denominator[BernoulliB[n]]; a[n_ /; Mod[n, 4] == 1] = 2; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2014 *)

Formula

a(2n) = A002445(n), a(2n+1) = A000034(n+1).
Showing 1-4 of 4 results.

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