A164020 -id:A164020 - cp's OEIS frontend

A165299 a(n) = A002790(n) / A164020(n).

1, 1, 1, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 60, 60, 1, 3, 10, 10, 42, 42, 20, 12, 30, 30, 252, 36, 4, 4, 8, 8, 231, 231, 70, 210, 2, 2, 5460, 5460, 14, 42, 660, 132, 1260, 1260, 56, 840, 210, 210, 7956, 7956, 44, 396, 440, 440, 228, 228, 40, 120, 24, 24, 720720, 144144, 715, 2145, 102, 510

Offset: 0

Paul Curtz, Sep 14 2009

nonn

A002790 := proc(n) add((-1)^k*stirling1(n, k)/(k+1), k=0..n) ; denom(%) ; end proc:
A002445 := proc(n) bernoulli(2*n) ; denom(%) ; end proc:
A164020 := proc(n) if type(n,'even') then A002445(n/2) ; else n+1 ; end if; end proc:
A165299 := proc(n) A002790(n)/A164020(n) ; end proc:
seq(A165299(n),n=0..80) ; # R. J. Mathar, Jul 04 2011

A176327 Numerators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

Original entry on oeis.org

1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Paul Curtz, Apr 15 2010

sign

Numerator of the Bernoulli number B_n, except B(1)=0.
A027641 is the main entry for this sequence, which is only a minor variation. - N. J. A. Sloane, Nov 29 2010.
This could formally be defined by building the arithmetic mean of the numerators in A164555(n) and A027641(n).

Antti Karttunen, Table of n, a(n) for n = 0..200

Cf. A176289 (denominators), A027642, A141056, A164020, A165823

seq(numer((bernoulli(i,0)+bernoulli(i,1))/2),i=0..40); # Peter Luschny, Jun 17 2012

terms = 41; egf = (x/2)*((1 + Exp[-x])/(1 - Exp[-x])) + O[x]^(terms+1);
CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* Jean-François Alcover, Jun 13 2017 *)

apply(numerator, Vec(serlaplace((x/2)*(1+exp(-x))/(1-exp(-x))))) \\ Charles R Greathouse IV, Sep 26 2017

a(2n+1) = 0. a(2n ) = A000367(n).

a(n) = A164555(n) = A027641(n) if n <>1.

New name from Peter Luschny, Jun 18 2012

A181722 Numerator of (1/n - Bernoulli number A164555(n)/A027642(n)).

Original entry on oeis.org

0, 0, 1, 1, 7, 1, 5, 1, 13, 1, 1, 1, 901, 1, -11, 1, 3647, 1, -43825, 1, 1222387, 1, -854507, 1, 1181821001, 1, -76977925, 1, 23749461059, 1, -8615841275543, 1, 28267510484519, 1

Offset: 1

Paul Curtz, Nov 17 2010

An autosequence is a sequence whose inverse binomial transform is the sequence signed. In integers, the oldest example is Fibonacci A000045. In fractions, A164555/A027642 is the son of 1/n via the Akiyama-Tanigawa algorithm; grandson is (A174110/A174111) = 1/2, 2/3, 1/2, 2/15, ...; see A164020. See A174341/A174342. All are from the same family.

			Fractions are 0, 0, 1/6, 1/4, 7/30, 1/6, 5/42, 1/8, 13/90, 1/10, 1/66, 1/12, 901/2730, ...

G. C. Greubel, Table of n, a(n) for n = 1..625
OEIS Wiki, Autosequence.

Cf. A000045, A027642, A164020, A164555, A174110, A174111, A174341, A174342.

A181722:= func< n | n le 2 select 0 else Numerator(1/n - Bernoulli(n-1)) >;
[A181722(n): n in [1..40]]; // G. C. Greubel, Mar 25 2024

a[n_] := If[n <= 2, 0, Numerator[1/n - BernoulliB[n-1]]];
Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Jun 07 2017 *)

def A181722(n): return 0 if n<3 else numerator(1/n - bernoulli(n-1))
[A181722(n) for n in range(1,41)] # G. C. Greubel, Mar 25 2024

A176144 a(2n) = A164555(n). a(2n+1) = A027641(n).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 5, 5, 0, 0, -691, -691, 0, 0, 7, 7, 0, 0, -3617, -3617, 0, 0, 43867, 43867, 0, 0, -174611, -174611, 0, 0, 854513, 854513, 0, 0, -236364091, -236364091, 0, 0, 8553103, 8553103, 0, 0, -23749461029, -23749461029, 0, 0, 8615841276005

Offset: 0

Paul Curtz, Apr 10 2010

Formally, these are the numerators of a sequence of fractions defined by alternating A164555(n)/A027642(n) with A027641(n)/A027642(n),
which apart from the third term duplicates the Bernoulli numbers.
Essentially a duplication of the entries of A027641.

Cf. A141056, A164020, A141056.

Edited by R. J. Mathar, Jun 07 2010

A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

Original entry on oeis.org

1, 2, 6, 24, 30, 1440, 42, 120960, 30, 7257600, 66, 958003200, 2730, 5230697472000, 6, 62768369664000, 510, 64023737057280000

Offset: 0

Paul Curtz, Sep 28 2009

b(n)=a(2n+1)/a(2n) =2,4,48,2880,241920,145152,= 2*(1,2,24,1440,=1,2*A141421). Among other denominators, A027642,A141056,A164020. 2*A141421 is second bisection of A091137 which is linked to Bernoulli via A027760. See A160014,von Staudt-Clausen theorem.

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).

cp's OEIS Frontend

A165299 a(n) = A002790(n) / A164020(n).

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A176327 Numerators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

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A181722 Numerator of (1/n - Bernoulli number A164555(n)/A027642(n)).

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A176144 a(2n) = A164555(n). a(2n+1) = A027641(n).

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A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

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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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