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A090801 List of distinct numbers appearing as denominators of Bernoulli numbers. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A241557 A006954 A265501 * A166062 A100194 A229882

Adjacent sequences:  A090798 A090799 A090800 * A090802 A090803 A090804

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Extended by Robert G. Wilson v, Feb 10 2004

STATUS

approved

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Last modified March 27 12:14 EDT 2017. Contains 284176 sequences.