A376485 Carmichael numbers ordered by largest prime factor, then by size.

561, 1105, 1729, 2465, 2821, 75361, 63973, 1050985, 6601, 41041, 29341, 172081, 552721, 852841, 10877581, 1256855041, 8911, 340561, 15182481601, 72720130561, 10585, 15841, 126217, 825265, 2433601, 496050841, 672389641, 5394826801, 24465723528961, 1074363265, 24172484701, 62745, 2806205689, 22541365441, 46657, 2113921, 6436473121, 6557296321, 13402361281, 26242929505, 65320532641, 143873352001, 105083995864811041

Offset: 1

Author

Charles R Greathouse IV, Sep 24 2024

Keywords

• nonn
• tabf

Examples

			17: 561, 1105;
19: 1729;
23:
29: 2465;
31: 2821, 75361;
37: 63973, 1050985;
41: 6601, 41041;
43:
47:
53:
59:
61: 29341, 172081, 552721, 852841, 10877581, 1256855041;
67: 8911, 340561, 15182481601;
71: 72720130561;
73: 10585, 15841, 126217, 825265, 2433601, 496050841, 672389641, 5394826801, 24465723528961;
79: 1074363265, 24172484701
83:
89: 62745, 2806205689, 22541365441;
97: 46657, 2113921, 6436473121, 6557296321, 13402361281, 26242929505, 65320532641, 143873352001, 105083995864811041
101: 101101, 252601, 2100901, 9494101, 6820479601, 109038862801, 102967089120001
		

Crossrefs

Cf. A002997, A081702, A283715 (row lengths).

Programs

PARI

\\ This program is inefficient and functions as proof-of-concept only.
Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
car(n)=n%2 && !isprime(n) && Korselt(n) && n>1
row(k)=my(p=prime(k)); fordiv(prod(i=2,k-1,prime(i)),n,if(car(p*n), print1(p*n,", ")))

Python

from itertools import islice, combinations
from math import prod
from sympy import nextprime
def A376485_gen(): # generator of terms
    plist, p = [3, 5], 7
    while True:
        clist = []
        for l in range(2,len(plist)+1):
            for q in combinations(plist,l):
                k = prod(q)*p-1
                if not (k%(p-1) or any(k%(r-1) for r in q)):
                    clist.append(k+1)
        yield from sorted(clist)
        plist.append(p)
        p = nextprime(p)
A376485_list = list(islice(A376485_gen(),43)) # Chai Wah Wu, Sep 25 2024