

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
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OFFSET

1,2


COMMENTS

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 p1 for which p is a prime divisor of k.
Now list the primes p such that p1 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 21, 31 and 131 is 12, so 2n=12. The primes p such that p1 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)
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



FORMULA

We know from the von StaudtClausen theorem (see Rademacher) that the denominator of the Bernoulli number B_{2k} is the product of those distinct primes p for which p1 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



PROG

(PARI) is(n)=if(n==1, 1, my(f=factor(n)); if(vecmax(f[, 2])>1, return(0)); fordiv(lcm(apply(k>k1, f[, 1])), k, if(isprime(k+1) && n%(k+1), return(0))); 1) \\ Charles R Greathouse IV, Nov 26 2012


CROSSREFS



KEYWORD

nonn,easy


AUTHOR

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


EXTENSIONS



STATUS

approved



