A182515 Carmichael numbers of the form C = 23*67*(66n+23).

340561, 9494101, 499310197, 717164449, 1330655041, 1831048561, 2586927553, 2806205689, 3088134721, 3284630713, 5394826801, 5447713921, 6150705793, 7349616121, 10501586767, 11453249809, 12820178449, 13714377601, 13968642601, 15818878153, 23196631393, 23392517149, 26242929505

Offset

1,1

Comments

We get Carmichael numbers for n = 3, 93, 4909, 7051, 13083, 18003, 25435, 27591, 30363, 32295, 53043, 53563, 60475, 72263, 103254, 112611, 126051, 134843, 137343, 155535, 228075, 230001, 258027.

Conjecture: Any Carmichael number C divisible by 23 and 67 can be written as C = 23*67*(66n+23).

Checked for the first 23 Carmichael numbers divisible by 23 and 67.

Note: the possibility to can be written as C = a*b*(n*(b-1) +a), where a and b prime divisors of C, is a property of only some of Carmichael numbers, thus is not derived from Korselt's criterion (for instance, for the Carmichael number 29341 = 13*37*61, we have 61 mod 36 = 25). In fact, seems to be rather a property of some pairs of primes (other pair of primes which generates Carmichael numbers of this form is 41 and 241).

The conjecture follows from Korselt's criterion: 67 | a(n) so a(n) = 1 (mod 66). - Charles R Greathouse IV, Oct 02 2012

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

E. W. Weisstein, Carmichael Number

Prog

(PARI) 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
list(lim)=my(v=List()); forstep(n=340561, lim, 101706, if(Korselt(n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jul 05 2017

Crossrefs

Sequence in context: A103808 A251996 A182132 * A236703 A178900 A159462

Adjacent sequences: A182512 A182513 A182514 * A182516 A182517 A182518

Keyword

nonn

Author

Marius Coman, May 03 2012

Status

approved