A090801 List of distinct numbers appearing as denominators of Bernoulli numbers.

1, 2, 6, 30, 42, 66, 138, 282, 330, 354, 498, 510, 642, 690, 798, 870, 1002, 1074, 1362, 1410, 1434, 1518, 1578, 1590, 1770, 1806, 2082, 2154, 2298, 2478, 2490, 2658, 2730, 2802, 2874, 3018, 3102, 3210, 3318, 3378, 3486, 3522, 3882, 3894, 3954, 4110, 4314

Offset: 1

Author

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 10 2004

Keywords

• nonn
• easy

Comments

From Dean Hickerson, Oct 19 2007: (Start)

Except for a(0)=1, all denominators in A002445 are divisible by 6 and are squarefree. To test such a number k to see if it's in the sequence, let 2n be the least common multiple of all p-1 for which p is a prime divisor of k.

Now list the primes p such that p-1 divides 2n. If all of them are divisors of k, then k is in the sequence; otherwise it's not.

For example, consider k = 78 = 2 * 3 * 13. The LCM of 2-1, 3-1 and 13-1 is 12, so 2n=12. The primes p such that p-1 divides 12 are 2, 3, 5, 7 and 13. Since 5 and 7 are not divisors of 78, 78 is not in the sequence. (End)

From Paul Curtz, Oct 19 2012: (Start)

a(n+3) mod 9 = 6,3,6,3,3,3,6,3,3,6,3,6,6,6,.... (Also a(n+3) in base 9 mod 10.)

(a(n+2)-2)/4 = 0, 1, 7, 10, 16, 34, 70, 82, 88, 124, .... See A002445.

(a(n+4) - a(n+3))/12 = 2, 1, 3, 6, 12, 4, 2, 12, 1, 11, .... Is this always an integer? (End)

References

• H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1001

Crossrefs

Cf. A090810, A002445 (denominators of Bernoulli numbers B_2n).

Programs

Mathematica

Take[Union@Table[Denominator[BernoulliB[k]], {k, 0, 2000}], 80] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

PARI

is(n)=if(n==1, 1, 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

Formula

We know from the von Staudt-Clausen theorem (see Rademacher) that the denominator of the Bernoulli number B_{2k} is the product of those distinct primes p for which p-1 divides 2k. In particular, all numbers after the first two (which are the denominators of B_0 and B_1) are divisible by 6. - N. J. A. Sloane, Feb 10 2004

Extensions

Extended by Robert G. Wilson v, Feb 10 2004