OFFSET
1,1
COMMENTS
The quoted part of name indicates the small difference to Euclid numbers A006862. With a(n)-1 having k prime factors, by definition A006862(k)-1 divides a(n)-1. All Euclid-Carmichael numbers below 10^24 are listed. A Carmichael number n is odd, so 2 is always a prime factor of n-1. 2^m+1 is never a Carmichael number, so n-1 has at least two prime factors. a(n)-1 has 2,2,5,6,3,5,5,6,6,5,6,7,3,6,5,8,4,4,4,9,6,11,10,13 prime factors for the listed numbers. There is no Euclid-Carmichael number n up to 10^27 with n-1 having 12 prime factors.
There is no Euclid-Carmichael number m up to 2.3*10^26 with 14 or more prime factors of m-1. (A006862(14)-1) divides m-1 for (non Euclid-)Carmichael number m=230032113277661217835472161 [= 1 + (A006862(14)-1) * 2^4 * 3^4 * 19 * 673 * 1061]. - Hermann Stamm-Wilbrandt, Mar 23 2026
LINKS
Hermann Stamm-Wilbrandt, Carmichael numbers up to 10^24 converted and stored binary (11.45GB) for PARI/GP fast read, from Jonathan Webster's website.
EXAMPLE
5031181 is a term because 5031181 = 1 + 2^2 * 3^3 * 5 * 7 * 11^3 and the prime factors of 5031181-1 are the first five primes.
PROG
(PARI) \\ Uses file from Stamm-Wilbrandt Link above
b=read("carm10e24.bin"); foreach(b, c, f=factorint(c-1); k=#f[, 1]; if(f[k, 1]==prime(k), print1(c", ")))
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import nextprime, isprime, factorint
def A393389_gen(): # generator of terms
h, hset = [(1, (1, ))], {1}
while True:
m, ps = heappop(h)
if not isprime(m+1):
fs = factorint(m+1)
if max(fs.values()) == 1 and not any(m % (p-1) for p in fs):
yield m+1
for p in ps:
mp = m*p
if mp not in hset:
heappush(h, (mp, ps))
hset.add(mp)
q = nextprime(max(ps, default=1))
mp = m*q
if mp not in hset:
heappush(h, (mp, (ps+(q, ))))
hset.add(mp)
CROSSREFS
KEYWORD
hard,nonn
AUTHOR
Hermann Stamm-Wilbrandt, Mar 13 2026
STATUS
approved