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

1,1

Comments

Wright shows that this sequence is infinite. - Charles R Greathouse IV, Aug 05 2022

Wright's proof is based on the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression. - 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.

Mathematica

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

Prog

(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

Crossrefs

Subsequence of A002997.

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

Sequence in context: A309235 A104016 A002997 * A087788 A173703 A306338

Adjacent sequences: A355036 A355037 A355038 * A355040 A355041 A355042

Keyword

nonn

Author

Michel Marcus, Jun 16 2022

Status

approved