A112428 -id:A112428 - cp's OEIS frontend

A338443 Carmichael numbers with 11 prime factors.

60977817398996785, 105083995864811041, 107473646345582881, 132819104923908481, 145671955835893201, 161802381510126721, 165167398073764801, 206063729626916161, 263076030916096321, 292433912163313921, 292561243007134465, 337365329710615921, 388219799621120545

Offset: 1

Tim Johannes Ohrtmann, Oct 28 2020

nonn

			60977817398996785 = 5*7*17*19*23*37*53*73*79*89*233 and 4, 6, 16, 18, 22, 36, 52, 72, 78, 88, 232 all divide 60977817398996784.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..1142 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Tables relating to Carmichael numbers.

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338442 (Carmichael numbers with 3-10 prime factors).

PARI
```
is(n)={omega(n)==11&&is_A002997(n)}
```

Equals A002997 intersect A069272.

A355039 Carmichael numbers whose number of prime factors is prime.

Original entry on oeis.org

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

Michel Marcus, Jun 16 2022

nonn

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

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.

Subsequence of A002997.

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

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

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);

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

cp's OEIS Frontend

A338443 Carmichael numbers with 11 prime factors.

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