A248614 -id:A248614 - cp's OEIS frontend

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

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

Paul Curtz, Oct 16 2014

nonn

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, ... .

Michel Marcus, Table of n, a(n) for n = 0..842
Index entries for sequences related to Bernoulli numbers

Cf. A027642, A090126, A248614.

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

More terms from Michel Marcus, Nov 08 2014

A346468 a(n) = (n-1) / A346467(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 2, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 1, 21, 1, 23, 2, 25, 13, 27, 1, 29, 1, 31, 2, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 1, 45, 1, 47, 1, 49, 5, 51, 1, 53, 3, 55, 2, 57, 1, 59, 1, 61, 31, 63, 4, 65, 1, 67, 17, 69, 1, 71, 1, 73, 37, 75, 19, 77, 1, 79, 1, 81, 1, 83, 1, 85, 43, 87, 1, 89

Offset: 1

Antti Karttunen and Thomas Ordowski, Jul 22 2021

nonn

Numbers n such that a(n) = 1 are A248614(m)+1 for m > 0. These are all primes together with A317210. The set of these numbers has zero asymptotic density.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002322, A027642, A248614, A317210, A346467.

{0}~Join~Array[#/CarmichaelLambda@ Denominator@ BernoulliB@ # &, 89] (* Michael De Vlieger, Nov 23 2021 *)

A346468(n) = if(1==n,0,my(m=1); fordiv(n-1,d,if(isprime(1+d),m = lcm(m,d))); ((n-1)/m));

a(n) = (n-1) / A346467(n).

a(n) = (n-1) / A002322(A027642(n-1)).

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

Paul Curtz, Oct 28 2014

nonn

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

Index entries for sequences related to Bernoulli numbers

A variant of the Clausen numbers A141056, A160014. And of A176591.

Cf. A000034, A002445, A016813, A027642, A051222, A051226, A051228, A051230, A090126, A164020, A248614, A249134.

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

a[n_] := Denominator[BernoulliB[n]]; a[n_ /; Mod[n, 4] == 1] = 2; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2014 *)

a(2n) = A002445(n), a(2n+1) = A000034(n+1).

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

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A346468 a(n) = (n-1) / A346467(n).

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

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