A074379 Carmichael numbers with exactly 4 prime factors.

41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561, 449065, 552721, 656601, 658801, 670033, 748657, 838201, 852841, 997633, 1033669, 1082809, 1569457, 1773289, 2100901, 2113921, 2433601, 2455921

Offset: 1

Author

Jani Melik, Sep 24 2002

Keywords

• nonn

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.

Examples

			41041 = 7 * 11 * 13 * 41.
62745 = 3 * 5 * 47 * 89.
		

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

Crossrefs

Cf. A002997 (Carmichael numbers), A006931 (least Carmichael with n prime factors), A046386 (products of four distinct primes).

Programs

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} & ]

PARI

is_A074379(n)=is_A002997(n) && is_A046386(n) \\ M. F. Hasler, Mar 24 2022

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

Formula

Intersection of A002997 (Carmichael numbers) and A046386 (product of four distinct primes). - M. F. Hasler, Mar 24 2022

Extensions

Edited and extended by Robert G. Wilson v, Oct 03 2002

Edited by M. F. Hasler, Mar 24 2022