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A090801
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List of distinct numbers appearing as denominators of Bernoulli numbers.
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12
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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
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1,2
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COMMENTS
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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)
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)
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REFERENCES
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H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
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LINKS
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FORMULA
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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
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 10 2004
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EXTENSIONS
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STATUS
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approved
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