OFFSET
1,1
COMMENTS
Original name was: "Super-Carmichael numbers with exactly 4 factors", and a comment explained that the prefix "super" means that the Moebius function (A008683) equals mu(N) = +1 for these. But for squarefree numbers such as Carmichael numbers (A002997), this just means that they have an even number of prime factors, which is trivial if that number is 4.
In the literature there are other definitions of "super-Carmichael numbers", see the McIntosh and Meštrović references, so we prefer not to use this terminology at all.
LINKS
R. J. Mathar and Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..6042 from R. J. Mathar)
Richard J. McIntosh Carmichael numbers with (p + 1) | (n - 1), Integers 14 (2014) #A59.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867, May 04 2013
FORMULA
Intersection of A002997 (Carmichael numbers) and A046386 (product of four distinct primes). - M. F. Hasler, Mar 24 2022
EXAMPLE
41041 = 7 * 11 * 13 * 41.
62745 = 3 * 5 * 47 * 89.
MATHEMATICA
p = Table[ Prime[i], {i, 1, 10}]; f[n_] := Union[ PowerMod[ Select[p, GCD[ #, n] == 1 & ], n - 1, n]]; Select[ Range[2500000], !PrimeQ[ # ] && OddQ[ # ] && Length[ FactorInteger[ # ]] == 4 && MoebiusMu[ # ] == 1 && f[ # ] == {1} & ]
PROG
(PARI) list(lim)=my(v=List()); forprime(p=3, sqrtnint(lim\=1, 4), forprime(q=p+2, sqrtnint(lim\p, 3), if(q%p==1, next); forprime(r=q+2, sqrtint(lim\p\q), if(r%p==1 || r%q==1, next); my(m=lcm([p-1, q-1, r-1]), pqr=p*q*r, t=Mod(1, m)/pqr, L=lim\pqr); fordiv(pqr-1, d, my(s=d+1); if(s>L, break); if(s==t && s>r && isprime(s), listput(v, pqr*s)))))); Set(v) \\ Charles R Greathouse IV, Apr 23 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Jani Melik, Sep 24 2002
EXTENSIONS
Edited and extended by Robert G. Wilson v, Oct 03 2002
Edited by M. F. Hasler, Mar 24 2022
STATUS
approved