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A090801 List of distinct numbers appearing as denominators of Bernoulli numbers. 12

%I

%S 1,2,6,30,42,66,138,282,330,354,498,510,642,690,798,870,1002,1074,

%T 1362,1410,1434,1518,1578,1590,1770,1806,2082,2154,2298,2478,2490,

%U 2658,2730,2802,2874,3018,3102,3210,3318,3378,3486,3522,3882,3894,3954,4110,4314

%N List of distinct numbers appearing as denominators of Bernoulli numbers.

%C From _Dean Hickerson_, Oct 19 2007: (Start)

%C 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.

%C 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.

%C 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)

%C From _Paul Curtz_, Oct 19 2012: (Start)

%C 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.)

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

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

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

%H T. D. Noe, <a href="/A090801/b090801.txt">Table of n, a(n) for n = 1..1001</a>

%F 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

%t Take[Union@Table[Denominator[BernoulliB[k]], {k, 0, 2000}], 80] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)

%o (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

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

%K nonn,easy

%O 1,2

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

%E Extended by _Robert G. Wilson v_, Feb 10 2004

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Last modified August 1 17:38 EDT 2021. Contains 346402 sequences. (Running on oeis4.)