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

2,1

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.

Example

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

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 *)

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
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

Crossrefs

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

Sequence in context: A137198 A194231 A141705 * A290486 A253595 A047713

Adjacent sequences: A135718 A135719 A135720 * A135722 A135723 A135724

Keyword

nonn,changed

Author

Artur Jasinski, Nov 25 2007

Extensions

More terms from Michel Marcus, Nov 07 2013

Escape clause added by Jianing Song, Dec 12 2021

Status

approved