OFFSET
1,1
COMMENTS
The number (7m+1)(8m+1)(11m+1) is a 3-factor Carmichael number if and only if m is equal to 1848k+942 with k in this sequence. The sequence includes the value k = 10^329 - 4624879 which yields a 1000-digit Carmichael number with three prime factors of 334 digits each. Other Carmichael numbers of the same form would necessarily have 4 prime factors or more; the smallest such example is 3664585=127*(7*29)*199, for m=18.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Gérard P. Michon, Generic Carmichael Numbers.
EXAMPLE
a(1)=13 because k=13 corresponds to m=24966, which yields a product of three primes (7m+1)(8m+1)(11m+1) equal to the Carmichael number 9585921133193329. (Among all Carmichael numbers with 16 or fewer digits, as first listed by Richard G. E. Pinch, this one features the largest "least prime factor".)
MAPLE
filter:= proc(n) local m;
m:= 1848*n+942;
andmap(isprime, [7*m+1, 8*m+1, 11*m+1])
end proc:
select(filter, [$1..10000]); # Robert Israel, May 14 2019
MATHEMATICA
q[k_] := Module[{m = 1848*k + 942}, PrimeQ[7*m + 1] && PrimeQ[8*m + 1] && PrimeQ[11*m + 1]]; Select[Range[6000], q] (* Amiram Eldar, Apr 27 2024 *)
PROG
(Magma) [k:k in [1..5100]| forall{s:s in [7, 8, 11]|IsPrime(m*s+1) where m is 1848*k+942}]; // Marius A. Burtea, Nov 01 2019
(PARI) is(k) = {my(m = 1848*k + 942); isprime(7*m + 1) && isprime(8*m + 1) && isprime(11*m + 1); } \\ Amiram Eldar, Apr 27 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gerard P. Michon, Dec 03 2004
STATUS
approved