A002445 -id:A002445 - cp's OEIS frontend

A193267 The number 1 alternating with the numbers A006953/A002445 (which are integers).

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 2, 1, 24, 1, 2, 1, 4, 1, 6, 1, 32, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 80, 1, 2, 1, 84, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 96, 1, 2, 1, 100

Offset: 1

Paul Curtz, Dec 20 2012

nonn

a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing n and p-1 divides n. - Peter Luschny, Mar 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

using Nemo
function A193267(n) P = 1
    for (p, e) in factor(ZZ(n))
        divisible(ZZ(n), p - 1) && (P *= p^e) end
P end
[A193267(n) for n in 1:100] |> println # Peter Luschny, Mar 12 2018

[Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

with(numtheory); a := proc(n) divisors(n); map(i->i+1, %); select(isprime, %);
mul(k^padic[ordp](n,k),k=%) end: seq(a(n), n=1..100); # Peter Luschny, Mar 12 2018
# Alternatively:
A193267 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(n)[2]; for f in F do if divides(f[1]-1, n) then
P := P*f[1]^f[2] fi od; P end: seq(A193267(n), n=1..100); # Peter Luschny, Mar 12 2018

a[n_] := If[OddQ[n], 1, Denominator[ BernoulliB[n]/n ] / Denominator[ BernoulliB[n]] ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 21 2012 *)

a(n+1) = A185633(n+1)/A027760(n+1).

a(n+1) = c(n+2)/c(n+1).

A138239 Triangle read by rows: T(n,k) = A000040(k) if A002445(n) mod A000040(k) = 0, otherwise 1.

Original entry on oeis.org

1, 2, 3, 2, 3, 5, 2, 3, 1, 7, 2, 3, 5, 1, 1, 2, 3, 1, 1, 11, 1, 2, 3, 5, 7, 1, 13, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 17, 1, 1, 2, 3, 1, 7, 1, 1, 1, 19, 1, 1, 2, 3, 5, 1, 11, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 2, 3, 5, 7, 1, 13, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Mats Granvik, Mar 07 2008

Row products give A002445.

A prime number appears in a column at every A130290-th row from the (A130290+1)-th row onwards. The prime numbers are, so to speak, equidistantly distributed in the columns. A130290 is essentially A005097. Counting terms > 1 in the rows gives A046886.

			First few rows of the triangle and row products are:
1 = 1
2*3 = 6
2*3*5 = 30
2*3*1*7 = 42
2*3*5*1*1 = 30
2*3*1*1*11*1 = 66
2*3*5*7*1*13*1 = 2730

Michel Marcus, Rows n=0..100 of triangle, flattened

Cf. A002445, A005097, A046886, A130290.

T:= (n, k)-> (p-> `if`(irem(denom(bernoulli(2*n)), p)=0, p, 1))(ithprime(k)):
seq(seq(T(n, k), k=1..n+1), n=0..20);  # Alois P. Heinz, Aug 27 2017

t[n_, k_] := If[Mod[Denominator[BernoulliB[2n]], (p = Prime[k])] == 0, p, 1];
Flatten[Table[t[n, k], {n, 0, 13}, {k, 1, n+1}]][[1 ;; 102]] (* Jean-François Alcover, Jun 16 2011 *)

tabl(nn) = {for (n=0, nn, dbn = denominator(bernfrac(2*n)); for (k=1, n+1, if (! (dbn % prime(k)), w = prime(k), w = 1); print1(w, ", "); ); print; ); } \\ Michel Marcus, Aug 27 2017

Definition edited by N. J. A. Sloane, Mar 18 2010

Offset corrected by Alois P. Heinz, Aug 27 2017

A073409 Largest prime factor of the denominator of the Bernoulli number B(2*n) (A002445).

Original entry on oeis.org

3, 5, 7, 5, 11, 13, 3, 17, 19, 11, 23, 13, 3, 29, 31, 17, 3, 37, 3, 41, 43, 23, 47, 17, 11, 53, 19, 29, 59, 61, 3, 17, 67, 5, 71, 73, 3, 5, 79, 41, 83, 43, 3, 89, 31, 47, 3, 97, 3, 101, 103, 53, 107, 109, 23, 113, 7, 59, 3, 61, 3, 5, 127, 17, 131, 67, 3, 137, 139, 71, 3, 73, 3, 149

Offset: 1

Benoit Cloitre, Aug 23 2002

Least k such that k!*B(2n) is an integer where B(2n) denotes the 2n-th Bernoulli number.
a((p-1)/2) = p, where p is odd prime. All odd primes appear in this sequence. The very first appearance of odd prime p is a((p-1)/2). - Alexander Adamchuk, Jul 31 2006
Conjecture: a(n) is the largest prime p <= 2n+1 such that p * A000367(n) == - A002445(n) (mod p^2) for n > 0. Note that 2^(2n) == 1 (mod a(n)), since a(n) is the largest prime p such that b^(2n)== 1 (mod p) for every b coprime to p; i.e., a(n) is the largest prime p such that p-1 | 2n. - Thomas Ordowski, May 17 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000367, A002445, A005097.

Table[FactorInteger[Denominator[BernoulliB[2n]]][[ -1, 1]], {n, 100}]

a(n)=
{
    my(bd=1);
    forprime (p=2, 2*n+1, if( (2*n)%(p-1)==0, bd=p ) );
    return(bd);
}
/* Joerg Arndt, May 06 2012 */

a(n)=my(p); fordiv(n,d, if(isprime(p=2*n/d+1), return(p))) \\ Charles R Greathouse IV, Jun 08 2020

A177735 a(0)=1, a(n)=A002445(n)/6 for n>=1.

Original entry on oeis.org

1, 1, 5, 7, 5, 11, 455, 1, 85, 133, 55, 23, 455, 1, 145, 2387, 85, 1, 319865, 1, 2255, 301, 115, 47, 7735, 11, 265, 133, 145, 59, 9464455, 1, 85, 10787, 5, 781, 23350145, 1, 5, 553, 38335, 83, 567385, 1, 10235, 45353, 235, 1, 750295, 1, 5555, 721, 265, 107

Offset: 0

Paul Curtz, May 12 2010

nonn

For n>=1: denominators of the Bernoulli numbers (A002445) divided by 6.
All entries are odd.
a(n)= A002445(n) / A020793(n).
5 divides a(2*n) for n>=1.
These numbers also equal to the lengths of the repeating patterns for the excluded integer values of c/6, when both p^n + c and p^n - c are prime, for an infinite number of primes p>2, and a given integer n>0, arising from the union of one or more prime-based modulo cycles, determined by the divisors of n. See A005097 for details and connection to the von Staudt-Clausen Theorem below. - Richard R. Forberg, Jul 19 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993.
Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem

Cf. A002445, A165949.

A002445 := proc(n) bernoulli(2*n) ; denom(%) ; end proc:
A177735 := proc(n) if n = 0 then 1; else A002445(n)/6 ; end if; end proc:
seq(A177735(n),n=0..60) ; # R. J. Mathar, Aug 15 2010

Join[{1},Denominator[BernoulliB[Range[2,120,2]]]/6] (* Harvey P. Dale, Oct 19 2012 *)
result = {}; Do[prod = 1; Do[If[PrimeQ[2*Divisors[n][[i]] + 1], prod *= (2*Divisors[n][[i]] + 1)], {i, 2, Length[Divisors[n]]}];
AppendTo[result, prod] , {n, 1, 100}]  ; result (* Richard R. Forberg, Jul 19 2016 *)

a(n)=
{
    my(bd=1);
    forprime (p=5, 2*n+1, if( (2*n)%(p-1)==0, bd*=p ) );
    bd;
}
/* Joerg Arndt, May 06 2012 */

a(n)=if(n<2, return(1)); my(s=1); fordiv(n,d, if(isprime(2*d+1) && d>1, s *= 2*d+1)); s \\ Charles R Greathouse IV, Jul 20 2016

def A177735(n):
    if n == 0: return 1
    M = map(lambda i: i+1, divisors(2*n))
    return mul(filter(lambda s: is_prime(s), M))//6
print([A177735(n) for n in (0..53)]) # Peter Luschny, Feb 20 2016

a(n) = denominator(BernoulliB(2*n, 1/2))/(3*2^(2*n)). - Jean-François Alcover, Apr 16 2013

A simple direct calculation of the denominators, for n>=1, is based on the von Staudt-Clausen Theorem: Product{d|n}(2d+1), for d>1 and 2d+1 prime. See in the Mathematica section below. - Richard R. Forberg, Jul 19 2016

A140814 a(0)=3, a(n)=A002445(n) for n >= 1.

Original entry on oeis.org

3, 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

Offset: 0

Paul Curtz, Jul 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A027762, A140816.

[3] cat [Denominator(Bernoulli(2*n)): n in [1..60]]; // Vincenzo Librandi, Nov 04 2018

Join[{3}, Table[Denominator[BernoulliB[2 n]], {n, 60}]] (* Vincenzo Librandi, Nov 04 2018 *)

a(n) = A106458(2*n) + A106458(2*n+1).

a(n) = A027762(n) for n >= 1. - Georg Fischer, Nov 03 2018

Edited and extended by R. J. Mathar, Jul 29 2008

A035078 Numerators of partial sums of Bernoulli numbers B_{2n} = A000367/A002445.

Original entry on oeis.org

1, 7, 17, 81, 118, 2771, 4737, 63457, -1270924, 161636091, -464743285, 254905515589, -3006818262414, 299981806371451, -83955854172826681, 38482697321210434701, -1458143803622109300584, 83247435772128371635117

Offset: 0

N. J. A. Sloane

Index entries for sequences related to Bernoulli numbers.

A134825 Floor of the even-indexed Bernoulli numbers B_{2n} = A000367(n)/A002445(n).

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 1, -8, 54, -530, 6192, -86581, 1425517, -27298232, 601580873, -15116315768, 429614643061, -13711655205089, 488332318973593, -19296579341940069, 841693047573682615, -40338071854059455414, 2115074863808199160560, -120866265222965259346028

Offset: 0

Wolfdieter Lang, Nov 13 2007

			n=4: B_8=-1/30=-0,033... hence a(4)=-1.

C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall, 2006, p. 107.

Cf. A000367, A002445, A014509, A027641, A027642.

Floor@BernoulliB[2 Range[0, 20]] (* Vladimir Reshetnikov, Nov 12 2015 *)

vector(30, n, n--; floor(bernfrac(2*n))) \\ Altug Alkan, Nov 12 2015

a(n) = floor(B_(2n)), n>=0, with B_{2n} = A000367(n)/A002445(n) = A027641(2n)/A027642(2n).

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.

A180943 Odd composite numbers m for which 12|A000367((m+1)/2)|==(-1)^{(m-1)/ 2} A002445((m+1)/2) (mod m).

Original entry on oeis.org

33, 169, 481, 561, 793, 805, 949, 1105, 1261, 1417, 1645, 1729, 2041, 2353, 2465, 2509, 2821, 2977, 3133, 3421, 3445, 3601, 4069, 4123, 4381, 4537, 4849, 5161, 5317, 5473, 5629, 5941, 6061, 6205, 6601, 7033, 7093, 7189, 7501, 7813, 7885, 7969, 8113

Offset: 1

Vladimir Shevelev, Sep 27 2010

nonn

These are pseudoprimes in the sense that the congruence of the definition is valid if any odd prime is substituted for m.
Entries of the form m = 4*k+3 are apparently rare: 4123, 8911, ...
Computed to 50 terms by D. S. McNeil, Sep 05 2010.

V. Shevelev, B-pseudoprimes, seqfan list, Sep 04 2010
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. [_N. J. A. Sloane_, Feb 07 2013]

Cf. A000367, A002445, A180942.

A000367 := proc(n) numer(bernoulli(2*n)) ; end proc:
A002445 := proc(n) denom(bernoulli(2*n)) ; end proc:
isA180943 := proc(m) if type(m,'odd') and not isprime(m) then 12*abs(A000367((m+1)/2)) mod m = (-1)^((m-1)/2)*A002445((m+1)/2) mod m ; else false; end if; end proc:
A180943 := proc(n) option remember; if n = 1 then 33; else for a from procname(n-1)+2 by 2 do if isA180943(a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Oct 24 2010

nb[n_] := Numerator[BernoulliB[2n]];
db[n_] := Denominator[BernoulliB[2n]];
okQ[m_] := CompositeQ[m] && Mod[12*Abs[nb[(m+1)/2]], m] == Mod[(-1)^((m-1)/2)*db[(m+1)/2], m];
Select[Range[33, 9999, 2], okQ] (* Jean-François Alcover, Feb 28 2024 *)

Comments rephrased and program added by R. J. Mathar, Oct 24 2010

Typo in data fixed by Jean-François Alcover, Feb 28 2024

A228838 a(n) = n * A002445(n).

Original entry on oeis.org

0, 6, 60, 126, 120, 330, 16380, 42, 4080, 7182, 3300, 1518, 32760, 78, 12180, 214830, 8160, 102, 34545420, 114, 270600, 37926, 15180, 6486, 1113840, 1650, 41340, 21546, 24360, 10266, 1703601900, 186, 16320, 2135826, 1020, 164010, 5043631320, 222, 1140

Offset: 0

Paul Curtz, Sep 05 2013

nonn

a(n+1) is a multiple of A040031(n+1), sequence of period 2: 6, 12.
a(n) is divisible by A040879(n)=30 followed by the sequence of period 2: 6, 60. See A040214 and A165734.
Note that A164877(n) + A000367(n) = A164558(2n).

			a(0)=0*1, a(1)=1*6, a(2)=2*30=60,, a(3)=3*42=126.

Colin Barker, Table of n, a(n) for n = 0..1000

PARI
```
a(n)=n*denominator(bernfrac(2*n))
```

a(n) = A176328(2n) - A000367(n).
a(n) = A164877(n)/2.
a(n+1) = A111008(n) * A036283(n+1).
2*a(n) = A164558(2n) - A000367(n).
a(n) = A164558(2n) - A176328(2n).

Typo in data fixed by Colin Barker, Jul 03 2015

cp's OEIS Frontend

A193267 The number 1 alternating with the numbers A006953/A002445 (which are integers).

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A138239 Triangle read by rows: T(n,k) = A000040(k) if A002445(n) mod A000040(k) = 0, otherwise 1.

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A073409 Largest prime factor of the denominator of the Bernoulli number B(2*n) (A002445).

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A177735 a(0)=1, a(n)=A002445(n)/6 for n>=1.

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A140814 a(0)=3, a(n)=A002445(n) for n >= 1.

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A035078 Numerators of partial sums of Bernoulli numbers B_{2n} = A000367/A002445.

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A134825 Floor of the even-indexed Bernoulli numbers B_{2n} = A000367(n)/A002445(n).

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Mathematica

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

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A180943 Odd composite numbers m for which 12|A000367((m+1)/2)|==(-1)^{(m-1)/ 2} A002445((m+1)/2) (mod m).