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A324977 Denominator(Bernoulli_{m-1}) / m, where m is the n-th Carmichael number.

%I A324977 #24 Apr 24 2024 02:42:06
%S A324977 26805565070,76004922,702286000198710990,302278602666,5360679390,
%T A324977 423023231634556544606744470770,582934735516230690164248578,
%U A324977 106515855804560422705933720818,8763422623117673428800595536306232967379299351012370,9231375124608836430,94422948020637332890056101961518875879389605546105043450762033482730
%N A324977 Denominator(Bernoulli_{m-1}) / m, where m is the n-th Carmichael number.
%C A324977 a(n) is an integer, because an odd composite number m is a Carmichael number iff m divides the denominator of Bernoulli_{m-1} (by Korselt's criterion and the von Staudt-Clausen theorem). See Pomerance, Selfridge, & Wagstaff, page 1006, and Kellner & Sondow, section on Bernoulli numbers.
%H A324977 Amiram Eldar, <a href="/A324977/b324977.txt">Table of n, a(n) for n = 1..358</a>
%H A324977 Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; <a href="https://arxiv.org/abs/1902.10672">arXiv preprint</a>, arXiv:1902.10672 [math.NT], 2019-2021.
%H A324977 Carl Pomerance, John L. Selfridge, and Samuel S. Wagstaff, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0572872-7">The pseudoprimes to 25*10^9</a>, Math. Comp., 35 (1980), 1003-1026.
%H A324977 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%H A324977 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%F A324977 a(n) = A027642(A002997(n)-1)/A002997(n).
%e A324977 The 1st Carmichael number is 561, and the denominator of Bernoulli_560 is 15037922004270, so a(1) = 15037922004270 / 561 = 26805565070.
%p A324977 with(numtheory): A324977 := proc(n) local C, Fc;
%p A324977 if n = 1 or irem(n,2) = 0 or isprime(n) then return NULL fi;
%p A324977 Fc := select(isprime, map(i->i+1, divisors(n-1)));
%p A324977 C := mul(i, i=Fc); if irem(C, n) <> 0 then NULL else C/n fi end:
%p A324977 seq(A324977(n), n=1..40000); # _Peter Luschny_, May 21 2019
%t A324977 carnum = Cases[Range[1, 100000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
%t A324977 Table[Denominator[BernoulliB[m - 1]]/m, {m, carnum}]
%Y A324977 Cf. A027642, A002997.
%K A324977 nonn
%O A324977 1,1
%A A324977 _Jonathan Sondow_, Mar 28 2019