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

26805565070, 76004922, 702286000198710990, 302278602666, 5360679390, 423023231634556544606744470770, 582934735516230690164248578, 106515855804560422705933720818, 8763422623117673428800595536306232967379299351012370, 9231375124608836430, 94422948020637332890056101961518875879389605546105043450762033482730

Offset: 1

Author

Jonathan Sondow, Mar 28 2019

Keywords

• nonn

Comments

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.

Examples

			The 1st Carmichael number is 561, and the denominator of Bernoulli_560 is 15037922004270, so a(1) = 15037922004270 / 561 = 26805565070.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..358
• Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
• Carl Pomerance, John L. Selfridge, and Samuel S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Math. Comp., 35 (1980), 1003-1026.
• Index entries for sequences related to Bernoulli numbers.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A027642, A002997.

Programs

Maple

with(numtheory): A324977 := proc(n) local C, Fc;
if n = 1 or irem(n,2) = 0 or isprime(n) then return NULL fi;
Fc := select(isprime, map(i->i+1, divisors(n-1)));
C := mul(i, i=Fc); if irem(C, n) <> 0 then NULL else C/n fi end:
seq(A324977(n), n=1..40000); # Peter Luschny, May 21 2019

Mathematica

carnum = Cases[Range[1, 100000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
Table[Denominator[BernoulliB[m - 1]]/m, {m, carnum}]

Formula

a(n) = A027642(A002997(n)-1)/A002997(n).