A090801 -id:A090801 - cp's OEIS frontend

A002445 Denominators of Bernoulli numbers B_{2n}.

1, 6, 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, 6, 33330, 4326, 1590, 642, 209191710, 1518, 1671270, 42

Offset: 0

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.
Row products of A138239. - Mats Granvik, Mar 08 2008
Equals row products of even rows in triangle A143343. In triangle A080092, row products = denominators of B1, B2, B4, B6, ... . - Gary W. Adamson, Aug 09 2008
Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is shown in A028246. - Gary W. Adamson, Aug 09 2008
There is a relation between the Euler numbers E_n and the Bernoulli numbers B_{2*n}, for n>0, namely, B_{2*n} = A000367(n)/a(n) = ((-1)^n/(2*(1-2^{2*n}))) * Sum_{k = 0..n-1} (-1)^k*2^{2*k}*C(2*n,2*k)*A000364(n-k)*A000367(k)/a(k). (See Bucur, et al.) - L. Edson Jeffery, Sep 17 2012
a(n) is the product of all primes of the form (k + n)/(k - n). - Thomas Ordowski, Jul 24 2025

			B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, ... ].

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 136.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
See A000367 for further references and links (there are a lot).

T. D. Noe, Table of n, a(n) for n = 0..10000
Amelia Bucur, José Luis López-Bonilla, and Jaime Robles-García, A note on the Namias identity for Bernoulli numbers, Journal of Scientific Research (Banaras Hindu University, Varanasi), Vol. 56 (2012), 117-120.
Suyuong Choi and Younghan Yoon, A decomposition of graph a-numbers, arXiv:2508.06855 [math.CO], 2025. See p. 13.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
Shizuo Kaji, Toshiaki Maeno, Koji Nuida, and Yasuhide Numata, Polynomial Expressions of Carries in p-ary Arithmetics, arXiv preprint arXiv:1506.02742 [math.CO], 2015.
Takao Komatsu, Florian Luca, and Claudio de J. Pita Ruiz V. , A note on the denominators of Bernoulli numbers, Proc. Japan Acad., 90, Ser. A (2014), p. 71-72.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6
Hong-Mei Liu, Shu-Hua Qi, and Shu-Yan Ding, Some Recurrence Relations for Cauchy Numbers of the First Kind, JIS 13 (2010) # 10.3.8.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Niels Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.
Niels Erik Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
Simon Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]
Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 2.
Index entries for sequences related to Bernoulli numbers.

Cf. A090801 (distinct numbers appearing as denominators of Bernoulli numbers)
B_n gives A027641/A027642. See A027641 for full list of references, links, formulas, etc.
See A000367 for numerators. Cf. A027762, A027641, A027642, A002882, A003245, A127187, A127188, A138239, A028246, A143343, A080092, A001897, A277087.
Cf. A160014 for a generalization.

[Denominator(Bernoulli(2*n)): n in [0..60]]; // Vincenzo Librandi, Nov 16 2014

A002445 := n -> mul(i,i=select(isprime,map(i->i+1,numtheory[divisors] (2*n)))): seq(A002445(n),n=0..40); # Peter Luschny, Aug 09 2011
# Alternative
N:= 1000: # to get a(0) to a(N)
A:= Vector(N,2):
for p in select(isprime,[seq(2*i+1,i=1..N)]) do
  r:= (p-1)/2;
  for n from r to N by r do
    A[n]:= A[n]*p
  od
od:
1, seq(A[n],n=1..N); # Robert Israel, Nov 16 2014

Take[Denominator[BernoulliB[Range[0,100]]],{1,-1,2}] (* Harvey P. Dale, Oct 17 2011 *)

a(n)=prod(p=2,2*n+1,if(isprime(p),if((2*n)%(p-1),1,p),1)) \\ Benoit Cloitre

A002445(n,P=1)=forprime(p=2,1+n*=2,n%(p-1)||P*=p);P \\ M. F. Hasler, Jan 05 2016

a(n) = denominator(bernfrac(2*n)); \\ Michel Marcus, Jul 16 2021

def A002445(n):
    if n == 0:
        return 1
    M = (i + 1 for i in divisors(2 * n))
    return prod(s for s in M if is_prime(s))
[A002445(n) for n in (0..57)] # Peter Luschny, Feb 20 2016

E.g.f: x/(exp(x) - 1); take denominators of even powers.
B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/ (2*Pi)^(2*n).
If n>=3 is prime,then a((n+1)/2)==(-1)^((n-1)/2)*12*|A000367((n+1)/2)|(mod n). - Vladimir Shevelev, Sep 04 2010
a(n) = denominator(-I*(2*n)!/(Pi*(1-2*n))*integral(log(1-1/t)^(1-2*n) dt, t=0..1)). - Gerry Martens, May 17 2011
a(n) = 2*denominator((2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 28 2012
a(n) = gcd(2!S(2n+1,2),...,(2n+1)!S(2n+1,2n+1)). Here S(n,k) is the Stirling number of the second kind. See the paper of Komatsu et al. - Istvan Mezo, May 12 2016
a(n) = 2*A001897(n) = A027642(2*n) = 3*A277087(n) for n>0. - Jonathan Sondow, Dec 14 2016

A092307 Primes p such that there are no primes q, 3 < q < p, such that (q-1) divides (p-1).

Original entry on oeis.org

5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 239, 263, 347, 359, 383, 443, 467, 479, 503, 563, 587, 647, 659, 719, 827, 839, 863, 887, 983, 1019, 1187, 1223, 1259, 1283, 1307, 1319, 1367, 1439, 1487, 1499, 1523, 1619, 1787, 1823, 1847, 1907, 2027, 2039, 2063

Offset: 1

T. D. Noe, Feb 12 2004

nonn

Using a sieve, these primes can be generated quickly. In the set of primes < 10^9, the density of these primes is about 1/10. It is easy to show that this sequence contains all "safe" primes (A005385).
Primes p such that 6p is the denominator of some Bernoulli number. - T. D. Noe, Sep 26 2006
Except for 5 and 7, primes p such that 12p is the denominator of B(p - 1)/(p - 1) where B(n) is the Bernoulli number. [Peter Luschny, Dec 24 2008]
Primes p such that A027642(p-1) = 6p. Composites m such that A027642(m-1) = 6m are Carmichael numbers 310049210890163447, 18220439770979212619, ... - Amiram Eldar and Thomas Ordowski, May 26 2021

			11 is in the sequence because 10 is not a multiple of either 4 or 6.
13 is not in the sequence because, although 12 is not a multiple of 6 or 10, it is a multiple of 4.

T. D. Noe, Table of n, a(n) for n=1..1000
P. Luschny, Von Staudt prime number, definition and computation. [From _Peter Luschny_, Dec 24 2008]

Cf. A090801 (distinct numbers appearing as denominators of Bernoulli numbers)
Cf. A092308 (for p=prime(n), the number of primes q such that q-1 divides p-1).
Cf. A005385 (primes p such that (p-1)/2 is also prime).
Cf. A152951. [From Peter Luschny, Dec 24 2008]

For p>7: seq(`if`(denom(bernoulli(n-1)/(n-1))=12*n,n,NULL),n=2..500); # Peter Luschny, Dec 24 2008

t = Table[p = Prime[n]; Length[Select[Divisors[p - 1] + 1, PrimeQ]], {n, 311}]; Prime[Flatten[Position[t, 3]]]
npqQ[n_]:=NoneTrue[Prime[Range[3,PrimePi[n]-1]],Mod[n-1,#-1]==0&]; Select[ Prime[ Range[3,400]],npqQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 26 2019 *)

use ntheory ":all"; forprimes { say if (bernfrac($-1))[1] == 6*$ } 1000; # Dana Jacobsen, Dec 29 2015

use ntheory ":all"; forprimes { my $p=$; say if vecnone { $ > 3 && $ < $p-1 && is_prime($+1) } divisors($p-1); } 5,1000; # Dana Jacobsen, Dec 29 2015

Let h(x) = 12x(x + log(exp(-x) -1) - log(x)) and [x^n]S(h) denote the coefficient of x^n in the series expansion of h. Consider for n > 1 the relation n = denominator((n - 1)![x^n]S(h)). [Peter Luschny, Dec 24 2008]

A166062 a(n) = denominator(Bernoulli(prime(n) - 1)).

Original entry on oeis.org

2, 6, 30, 42, 66, 2730, 510, 798, 138, 870, 14322, 1919190, 13530, 1806, 282, 1590, 354, 56786730, 64722, 4686, 140100870, 3318, 498, 61410, 4501770, 33330, 4326, 642, 209191710, 1671270, 4357878, 8646, 4110, 274386, 4470, 2162622, 1794590070, 130074

Offset: 1

Paul Curtz, Oct 05 2009

nonn

Divisibility through terms of A008578 is a consequence of the Staudt-Clausen theorem.
(Vaguely similar divisibility properties are considered in A165248 and A165943.)
The first 250 entries are all different. Is this true in general?
Would sorting the entries yield the full A090801?
a(n) > 1 is the largest number k such that x*y^p == y*x^p (mod k) for all integers x and y, where p = prime(n). Example: x*y^19 == y*x^19 (mod 798). - Michel Lagneau, Apr 19 2012
Comment from Herbert Kociemba, May 29 2020: (Start)
For each n there is exactly one member of the sequence whose factorization has prime(n) as its largest prime factor, namely a(n). From this we conclude:
1. All elements of the sequence are different.
2. Not all denominators of Bernoulli numbers appear in this sequence. For example the denominator of B(20), 330=2*3*5*11 never appears because the unique sequence element with largest prime divisor 11=prime(5) is a(5)=2*3*11. (End)

Peter Luschny, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem.

Cf. A071772, A110936.

seq(denom(bernoulli(ithprime(n)-1)), n=1..38); # Peter Luschny, Jul 14 2019

Table[Denominator[BernoulliB[n - 1]], {n, Prime[Range[38]]}] (* Harvey P. Dale, Apr 22 2012 *)
Table[GCD @@ Table[(n^k - n), {n, 2, 13}], {k, Prime[Range[100]]}] (* Increase n to 80 and k to 1000 for first thousand terms. - Herbert Kociemba, May 05 2020 *)
a[i_] := Times @@ Select[Prime[Range[i]], Mod[Prime[i] - 1, # - 1] == 0&]; Table[a[i], {i, 1, 100}](* Herbert Kociemba, May 06 2020 *)

a(n)=denominator(bernfrac(prime(n)-1)) \\ Charles R Greathouse IV, Apr 30 2012

a(n) = A027642(A008578(n) - 1).

Edited by Peter Luschny, Jul 14 2019

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

Paul Curtz, Oct 09 2014

nonn

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

			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.

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

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 *)

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

A218755 Denominators of Bernoulli numbers which are == 6 (mod 9).

Original entry on oeis.org

6, 42, 330, 510, 690, 798, 870, 1410, 1518, 1590, 1770, 1806, 2490, 3102, 3210, 3318, 3894, 4110, 4326, 4470, 4686, 5010, 5190, 5370, 5478, 6486, 6810, 7062, 7890, 8070, 8142, 8646, 8790, 9366, 9510, 10410, 10770, 11022

Offset: 1

Paul Curtz, Nov 05 2012

The sequence contains the elements of A090801 which are == 6 (mod 9).
Conjecture: all first differences 36, 288, 180, 180,... of the sequence are multiples of 36.
The conjecture is true, since elements of A090801 are 2 mod 4. - Charles R Greathouse IV, Nov 22 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Second subset of the Bernoulli denominators: A090801 which are == 3 (mod 9).

Take[Union[Select[Denominator[BernoulliB[Range[1000]]],Mod[#,9]==6&]],60] (* Harvey P. Dale, Nov 28 2012 *)

is(n)=if(n%36-6, 0, my(f=factor(n)); if(vecmax(f[, 2])>1, return(0)); fordiv(lcm(apply(k->k-1, f[, 1])), k, if(isprime(k+1) && n%(k+1), return(0))); 1) \\ Charles R Greathouse IV, Nov 26 2012

a(n) is identical to A079148[n] up to a(14)=227. Most a(n) except 2,3,239,443,647,659,827,1223,1259,1499,1787... belong to A005385[n]: Safe primes p: (p-1)/2 is also prime.

Except for 2 and 3, the same as A092307. - T. D. Noe, Sep 25 2006

a(n) mod 9 appears to be always 1 or 7 (except the first term). Checked for the first 1000 terms.

a(n+2) - a(n+1) = 6, 3, 6, 18, 36, 12, 6, 36, 3, 33, 12, 27,... = (A090801(n+4) - A090801(n+3))/4 are multiples of 3.

cp's OEIS Frontend

A119660 Prime factor of the distinct numbers appearing as denominators of Bernoulli numbers A090801 that is greater than all previous a(n). a(1) = 2.

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A140958 A090801(2n-1)+A090801(2n).

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A218128 a(n) = (A090801(n+2) - 2)/4.

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A002445 Denominators of Bernoulli numbers B_{2n}.

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A092307 Primes p such that there are no primes q, 3 < q < p, such that (q-1) divides (p-1).

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A166062 a(n) = denominator(Bernoulli(prime(n) - 1)).

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

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A218755 Denominators of Bernoulli numbers which are == 6 (mod 9).

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