A355039 Carmichael numbers whose number of prime factors is prime.

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 46657, 52633, 115921, 162401, 252601, 294409, 314821, 334153, 399001, 410041, 488881, 512461, 530881, 825265, 1024651, 1050985, 1152271, 1193221, 1461241, 1615681, 1857241, 1909001, 2508013, 3057601, 3581761, 3828001

Offset: 1

Author

Michel Marcus, Jun 16 2022

Keywords

• nonn

Comments

Wright shows that this sequence is infinite on the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression. - Charles R Greathouse IV, Aug 05 2022, corrected by Amiram Eldar, Mar 25 2024

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10280 (terms below 10^13)
• D. R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Philos. Soc., Vol. 83, No. 1 (1978), pp. 357-375.
• Thomas Wright, Carmichael Numbers with Prime Numbers of Prime Factors, arXiv:2206.07254 [math.NT], 2022-2024.

Crossrefs

Subsequence of A002997.

Cf. A087788, A112428, A112430 (subsequences with 3, 5, 7 prime factors).

Programs

Mathematica

Select[Range[1, 10^6, 2], CompositeQ[#] && PrimeQ[PrimeNu[#]] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 16 2022 *)

PARI

pKorselt(m) = my(f=factor(m)); for(i=1, #f[, 1], if(f[i, 2]>1||(m-1)%(f[i, 1]-1), return(0))); #f~;
isok(m) = (m%2) && !isprime(m) && isprime(pKorselt(m)) && (m>1);

Python

from itertools import islice
from sympy import factorint, isprime, nextprime
def A355039_gen(): # generator of terms
    p, q = 3, 5
    while True:
        yield from (n for n in range(p+2,q,2) if max((f:=factorint(n)).values()) == 1 and not any((n-1) % (p-1) for p in f) and isprime(len(f)))
        p, q = q, nextprime(q)
A355039_list = list(islice(A355039_gen(),20)) # Chai Wah Wu, Jun 16 2022