OFFSET
1,1
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.
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.
EXAMPLE
The 1st Carmichael number is 561, and the denominator of Bernoulli_560 is 15037922004270, so a(1) = 15037922004270 / 561 = 26805565070.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Mar 28 2019
STATUS
approved