A006954 -id:A006954 - cp's OEIS frontend

A106458 Bernoulli number denominators, with zeros at the odd places.

1, 2, 6, 0, 30, 0, 42, 0, 30, 0, 66, 0, 2730, 0, 6, 0, 510, 0, 798, 0, 330, 0, 138, 0, 2730, 0, 6, 0, 870, 0, 14322, 0, 510, 0, 6, 0, 1919190, 0, 6, 0, 13530, 0, 1806, 0, 690, 0, 282, 0, 46410, 0, 66, 0, 1590, 0, 798, 0, 870, 0, 354, 0, 56786730

Offset: 0

Gary W. Adamson, May 02 2005

A027642 is the correct version of this sequence. - N. J. A. Sloane

Equals right border of triangle A159688 if zeros are inserted in A159688 to allow for (n+1) terms per row. - Gary W. Adamson, Apr 19 2009

			Solutions to the system of simultaneous equations with 5 rows: (-1/2, 1/6, 0, -1/30, 0).

Robert M. Young, "Excursions in Calculus" MAA, 1992, p. 91 J. H. Conway & R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 108

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A027642, A006954, A007318.

Cf. A159688. - Gary W. Adamson, Apr 19 2009

a[n_] := If[OddQ[n] && n>2, 0, BernoulliB[n] // Denominator]; Table[a[n],{n,0,60}] (* Jean-François Alcover, Dec 29 2012 *)
Join[{1,2},Riffle[BernoulliB[2*Range[50]]//Denominator,0]] (* Harvey P. Dale, Sep 29 2024 *)

A106458(n) = if((n%2)&&n>1,0,denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

In addition to generating functions as shown in A006954, the Bernoulli numbers starting with B(1) = -1/2 may be generated from the following system of simultaneous equations: (exemplified by 5 rows): 2 0 0 0 0 = -1 3 3 0 0 0 = -1 4 6 4 0 0 = -1 5 10 10 5 0 = -1 6 15 20 15 6 = -1.

Typo in one term corrected by Paul Curtz, Jul 16 2008

A165908 Irregular triangle with the terms in the Staudt-Clausen theorem for the nonzero Bernoulli numbers multiplied by the product of the associated primes.

Original entry on oeis.org

1, 2, -1, 6, -3, -2, 30, -15, -10, -6, 42, -21, -14, -6, 30, -15, -10, -6, 66, -33, -22, -6, 2730, -1365, -910, -546, -390, -210, 12, -3, -2, -3060, -255, -170, -102, -30, 44688, -399, -266, -114, -42

Offset: 0

Paul Curtz, Sep 30 2009

The decomposition of a nonzero Bernoulli number in the Staudt-Clausen format is B(n) = A000146(n) - sum_k 1/A080092(n,k) with a set of primes A080092 characterizing the right hand side.

If we multiply this equation by the product of the primes for a given n (which is in A002445), discard the left hand side, and list individually the terms associated with A000146 and each of the k, we get row n of the current triangle .

			The decomposition of B_10 is 5/66 = 1-1/2-1/3-1/11. Multiplied by the product 2*3*11=66 of the denominators this becomes 5=66-33-22-6, and the 4 terms on the right hand side become one row of the table.
1;
2,-1;
6,-3,-2;
30,-15,-10,-6;
42,-21,-14,-6;
30,-15,-10,-6;
66,-33,-22,-6;
2730,-1365,-910,-546,-390,-210;

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Cf. A000146, A165884, A006954 (first column).

A165908 := proc(n) local i,p; Ld := [] ; pp := 1 ; for i from 1 do p := ithprime(i) ; if (2*n) mod (p-1) = 0 then Ld := [op(Ld),-1/p] ; pp := pp*p ; elif p-1 > 2*n then break; end if; end do: Ld := [A000146(n),op(Ld)] ; [seq(op(i,Ld)*pp,i=1..nops(Ld))] ; end proc: # for n>=2, R. J. Mathar, Jul 08 2011

a146[n_] := Sum[ Boole[ PrimeQ[d+1]]/(d+1), {d, Divisors[2n]}] + BernoulliB[2n]; primes[n_] := Select[ Prime /@ Range[n+1], Divisible[2n, #-1]& ]; row[n_] := With[{pp = primes[n]}, Join[{a146[n]}, -1/pp]*Times @@ pp]; Join[{1}, Flatten[ Table[row[n], {n, 0, 9}]]] (* Jean-François Alcover_, Aug 09 2012 *)

Edited by R. J. Mathar, Jul 08 2011

A174289 Numerator of the n-th term of the inverse binomial transform of 1, 1/2, B_4, B_6, B_8,..., a modified Bernoulli sequence.

Original entry on oeis.org

1, -1, 1, -1, 11, -137, 4157, -44879, 74351, -23262859, 113428851, -18122193779, 593728889477, -17199344405209, 773610521462677, -398027397442098469, 7730820046943979149, -6072430937404995879629, 164713122370768078443379

Offset: 0

Paul Curtz, Mar 15 2010

The denominators are in A174276. The input sequence starts 1, 1/2, 1/6, -1/30, 1/42....

The inverse binomial transform generates 1, -1/2, 1/6, -1/30, 11/70, -137/210, 4157/2310,...

Cf. A000367, A164555, A006954.

read("transforms") ;
L := [1,1/2,seq(bernoulli(2*i),i=1..30)] ;BINOMIALi(L) ; apply(numer,%) ; # R. J. Mathar, Dec 02 2010

b[0]=1; b[1]=1/2; b[n_] := BernoulliB[2n-2]; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 18}] // Numerator (* Jean-François Alcover_, Aug 09 2012 *)

A165161 Numerator of the n-th term in the first differences of the binomial transform of the "original" Bernoulli numbers.

Original entry on oeis.org

1, 2, 5, 29, 31, 43, 41, 29, 31, 71, 61, 2039, 3421, 13, -1, -3107, 4127, 44665, -43069, -174281, 174941, 854651, -854375, -236361361, 236366821, 8553109, -8553097, -23749460159, 23749461899, 8615841290327

Offset: 0

Paul Curtz, Sep 06 2009

The binomial transform of the "original" Bernoulli numbers is 1, 3/2, 13/6, ... as mentioned in A164558.
The first differences of that sequence are 3/2 - 1 = 1/2, 13/6 - 3/2 = 2/3, 5/6, 29/30, 31/30, ... and the numerators of these differences are listed here.
The bisection a(2n) reappears (up to signs) as A162173(n+1).

Cf. A051717 (denominators), A164555, A027642.

read("transforms") :
A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; end if; end proc:
A027642 := proc(n) denom(bernoulli(n)) ; end proc:
nmax := 40:
BINOMIAL([seq(A164555(n)/A027642(n), n=0..nmax)]) :
map(numer,DIFF(%)) ; # R. J. Mathar, Jul 07 2011

a(2n) + A000367(n) = A006954(n+1) = A051717(2n+1).

a(2n+1) + a(2n+2) = A051717(2n+2) + A051717(2n+3), n > 0.

A132084 A051717(2n).

Original entry on oeis.org

1, 3, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6, 4501770

Offset: 0

Paul Curtz, Aug 26 2008

Essentially the same as A006954.

a(2n) + a(2n+1) = 4, 72, 96, 2736, 1308, 468, ... are multiples of 4.

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;
A051717 := proc(n) denom(C(n)) ; end: A132084 := proc(n) A051717(2*n) ; end: seq(A132084(n),n=0..120) ; # R. J. Mathar, Sep 07 2009

Extended by R. J. Mathar, Sep 07 2009

A165884 Irregular table of negated A080092 and a leading column of 1's.

Original entry on oeis.org

1, 1, -2, 1, -2, -3, 1, -2, -3, -5, 1, -2, -3, -7, 1, -2, -3, -5, 1, -2, -3, -11, 1, -2, -3, -5, -7, -13, 1, -2, -3, -1, -2, -3, -5, -17

Offset: 0

Paul Curtz, Sep 29 2009

The von Staudt-Clausen decomposition of nonzero Bernoulli numbers (see A164555 and A006954) states B(0)=1, B(1) = 1/2 = 1-1/2, B(2) = 1/6 = 1-1/2-1/3, B(4) = -1/30 = 1-1/2-1/3-1/5 etc.

We consider the denominators of the fractions in these sums, one sum per row. The first term in the sums is essentially the sequence of two 1's followed by A000146; this contributes a first column to this sequence here compared with table A080092.

			1;
1, -2;
1, -2, -3;
1, -2, -3, -5;
1, -2, -3, -7;
1, -2, -3, -5;
1, -2, -3, -11;
1, -2, -3, -5, -7, -13;
1, -2, -3;

Eric Weisstein, von Staudt-Clausen Theorem, MathWorld.

Cf. A046886 (row lengths minus 1), A000146.

A176546 Bernoulli numerators A000367 with an additional 1 inserted to represent B_1.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Apr 20 2010

Equivalent to adding a 1 in front of A000367, or removing zeros in A164555.
(One could also remove zeros in A027641 which would flip the sign of a(1)).
The denominators are in A006954.

			B_0=1/1, B_1=1/2 "originally", B_2=1/6, B_4=-1/30, B_6=1/42,...

Cf. A165908, A174276.

A214867 Quotients of (first) primorial numbers and denominators of Bernoulli numbers B 0, B 1, B 2, B 4, B 6,... .

Original entry on oeis.org

1, 1, 1, 1, 5, 77, 455, 187, 1616615, 437437, 8107385, 607759061, 53773464745, 111446982977, 2180460221945005, 706769865044243, 2275461421392965, 3770118333635711057, 19548063559901161830545, 4094603218587147211, 92990138354449826827902565

Offset: 0

Paul Curtz, Mar 10 2013

nonn

a(2*n+4) is divisible by 5 (because A006954(n+2)=6,30,42,30,... is divisible by A165734(n)=period of length 2: repeat 6,30).

			a(0) = 1/1, a(1)= 2/2, a(2) = 6/6, a(3) = 30/30, a(4) =210/42=5.
By product (see A080092):
1,
1,
1,
1,
5,
7  * 11,
5  *  7 *13,
11 * 17,
5  *  7 *11 *13 *17 *19,
7  * 11 *13 *19 *23,
5  * 11 *13 *17 *23 *29,
7  * 13 *17 *19 *23 *29 *31,
5  *  7 *11 *13 *17 *19 *29 *31 *37.

a[n_] := Product[ Prime[k], {k,1, n}] / Denominator[ BernoulliB[2*n-2] ]; a[0] = a[1] = 1; Table[a[n],{n,0,20}] (* Jean-François Alcover, Mar 15 2013 *)

a(n) = A002110(n)/A006954(n).

More terms from Jean-François Alcover, Mar 15 2013

A227127 The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,... old prime numbers). Reduced numerators of the second row.

Original entry on oeis.org

1, 1, 2, 8, 20, 12, 28, 16, 36, 60, 22, 72, 52, 28, 60, 96, 102, 36, 114, 80, 42, 132, 92, 144, 200, 104, 54, 112, 58, 120, 434, 128, 198, 68, 350, 72, 222, 228, 156, 240, 246, 84, 430, 88, 180, 92, 564, 576, 196, 100, 204, 312, 106, 540, 330, 336, 342, 116, 354, 240, 122

Offset: 0

Paul Curtz, Jul 02 2013

1/A008578(n) and successive rows:
1,          1/2,     1/3,     1/5,   1/7,
1/2,        1/3,     2/5,    8/35,           = c(n) = a(n)/b(n)
1/6,      -2/15,   18/35,
3/10,  -136/105,
67/42
b(n) is essentially A006094. See A209329.
a(n) yields to a permutation of A008578 (via 1, 1, 2, 8, 12, 16, 20, 22, ...): 1, 2, 3, 5, 11, 17, 7, 29,... .

			a(n) is the numerators of c(n): c(0) = 1-1/2 = 1/2, c(1) = 2*(1/2-1/3) = 1/3, c(2) = 3*(1/3-1/5) = 2/5, c(3) = 4*(1/5-1/7)=8/35.
a(3) = 4*2 = 8, a(4) = 5*4 = 20.

Cf. A002110, A006954.

a[0, 0] = 1; a[0, m_ /; m > 0] := 1/Prime[m]; a[n_, m_] := a[n, m] = (m+1)*(a[n-1, m ] - a[n-1, m+1]); Table[a[1, m] // Numerator, {m, 0, 60}] (* Jean-François Alcover, Jul 04 2013 *)

a(n) = (n+1)*A001223(n-1), for n>=3.

cp's OEIS Frontend

A106458 Bernoulli number denominators, with zeros at the odd places.

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A165908 Irregular triangle with the terms in the Staudt-Clausen theorem for the nonzero Bernoulli numbers multiplied by the product of the associated primes.

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A174289 Numerator of the n-th term of the inverse binomial transform of 1, 1/2, B_4, B_6, B_8,..., a modified Bernoulli sequence.

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A165161 Numerator of the n-th term in the first differences of the binomial transform of the "original" Bernoulli numbers.

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A132084 A051717(2n).

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A165884 Irregular table of negated A080092 and a leading column of 1's.

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A176546 Bernoulli numerators A000367 with an additional 1 inserted to represent B_1.

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A214867 Quotients of (first) primorial numbers and denominators of Bernoulli numbers B 0, B 1, B 2, B 4, B 6,... .

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A227127 The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,... old prime numbers). Reduced numerators of the second row.

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