login
Carmichael numbers ordered by largest prime factor, then by size.
0

%I #18 Sep 25 2024 15:40:48

%S 561,1105,1729,2465,2821,75361,63973,1050985,6601,41041,29341,172081,

%T 552721,852841,10877581,1256855041,8911,340561,15182481601,

%U 72720130561,10585,15841,126217,825265,2433601,496050841,672389641,5394826801,24465723528961,1074363265,24172484701,62745,2806205689,22541365441,46657,2113921,6436473121,6557296321,13402361281,26242929505,65320532641,143873352001,105083995864811041

%N Carmichael numbers ordered by largest prime factor, then by size.

%e 17: 561, 1105;

%e 19: 1729;

%e 23:

%e 29: 2465;

%e 31: 2821, 75361;

%e 37: 63973, 1050985;

%e 41: 6601, 41041;

%e 43:

%e 47:

%e 53:

%e 59:

%e 61: 29341, 172081, 552721, 852841, 10877581, 1256855041;

%e 67: 8911, 340561, 15182481601;

%e 71: 72720130561;

%e 73: 10585, 15841, 126217, 825265, 2433601, 496050841, 672389641, 5394826801, 24465723528961;

%e 79: 1074363265, 24172484701

%e 83:

%e 89: 62745, 2806205689, 22541365441;

%e 97: 46657, 2113921, 6436473121, 6557296321, 13402361281, 26242929505, 65320532641, 143873352001, 105083995864811041

%e 101: 101101, 252601, 2100901, 9494101, 6820479601, 109038862801, 102967089120001

%o (PARI) \\ This program is inefficient and functions as proof-of-concept only.

%o 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

%o car(n)=n%2 && !isprime(n) && Korselt(n) && n>1

%o row(k)=my(p=prime(k)); fordiv(prod(i=2,k-1,prime(i)),n,if(car(p*n), print1(p*n,", ")))

%o (Python)

%o from itertools import islice, combinations

%o from math import prod

%o from sympy import nextprime

%o def A376485_gen(): # generator of terms

%o plist, p = [3, 5], 7

%o while True:

%o clist = []

%o for l in range(2,len(plist)+1):

%o for q in combinations(plist,l):

%o k = prod(q)*p-1

%o if not (k%(p-1) or any(k%(r-1) for r in q)):

%o clist.append(k+1)

%o yield from sorted(clist)

%o plist.append(p)

%o p = nextprime(p)

%o A376485_list = list(islice(A376485_gen(),43)) # _Chai Wah Wu_, Sep 25 2024

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

%K nonn,tabf

%O 1,1

%A _Charles R Greathouse IV_, Sep 24 2024