A135721 a(n) is the smallest Carmichael number (A002997) divisible by the n-th prime, or 0 if no such number exists.

561, 1105, 1729, 561, 1105, 561, 1729, 6601, 2465, 2821, 29341, 6601, 334153, 62745, 2433601, 74165065, 29341, 8911, 10024561, 10585, 2508013, 55462177, 62745, 46657, 101101, 52633, 84350561, 188461, 278545, 1152271, 18307381, 410041, 2628073, 12261061, 838201

Offset: 2

Author

Artur Jasinski, Nov 25 2007

Keywords

• nonn

Examples

			561 is the first Carmichael number and its prime factors are 3, 11, 17 (2nd, 5th and 7th primes), so a(2), a(5) and a(7) are equal to 561. - _Michel Marcus_, Nov 07 2013
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10001 (calculated using data from Claude Goutier; terms 2..1000 from Donovan Johnson)
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Gérard P. Michon, Carmichael Multiples of Odd Primes.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A002997, A135717, A006931, A135719, A135720.

Programs

Mathematica

c = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n]; Table[First@ Select[c, Mod[#, Prime@ n] == 0 &], {n, 2, 16}] (* Michael De Vlieger, Aug 28 2015, after Artur Jasinski at A002997 *)

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
isA002997(n)=n%2 && !isprime(n) && Korselt(n) && n>1
a(n) = my(pn=prime(n),cn = 31*pn); until (isA002997(cn+=2*pn),); cn; \\ Michel Marcus, Nov 07 2013, improved by M. F. Hasler, Apr 14 2015

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
a(n,p=prime(n))=my(m=lift(Mod(1/p,p-1)),c=max(m,33)*p,mp=m*p); while(!isprime(c) && !Korselt(c), c+=mp); c \\ Charles R Greathouse IV, Apr 15 2015

Extensions

More terms from Michel Marcus, Nov 07 2013

Escape clause added by Jianing Song, Dec 12 2021