A290692 Carmichael numbers of the form p - 2 where p is a prime number.

561, 2465, 656601, 1909001, 174352641, 230996949, 275283401, 939947009, 1534274841, 3264820001, 5860426881, 6025532241, 25536531021, 36709177121, 53388707681, 54519328481, 56222911361, 101536702401, 105528976961, 180481509681, 196866607601, 239862350001, 329245587161, 347469383801, 347511324161

Offset: 1

Author

Altug Alkan, Aug 09 2017

Keywords

• nonn

Comments

Rotkiewicz mentioned the first six terms of this sequence at the end of page 59 of his article (Links section). But his list includes 2821 and 46657 (2823 = 3 * 941 and 46659 = 3 * 103 * 151), which should not be there.

Carmichael numbers of the form p + 2 where p is a prime number are 1105, 2821, 6601, 29341, 41041, 52633, ... (see also A272754 for corresponding prime numbers).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5901 (terms below 10^22 calculated using data from Claude Goutier; terms 1..591 from Robert Israel)
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• R. G. E. Pinch, Carmichael numbers up to 10^16, 10^16 to 10^17, 10^17 to 10^18
• Andrzej Rotkiewicz, On pseudoprimes having special forms and a solution of K. Szymiczek's problem, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A002997, A272754, A287591.

Programs

Maple

# Using data file from Richard Pinch
infile:= "carmichael-16": Res:= NULL;
do
  S:= readline(infile);
  if S = 0 then break fi;
  L:= sscanf(S,"%d");
  if nops(L) <> 1 then break fi;
  if isprime(L[1]+2) then Res:= Res, L[1]; fi
od:
Res; # Robert Israel, Jun 03 2019

Mathematica

Cases[Range[1, 10^7, 2], n_ /; And[Mod[n, CarmichaelLambda@ n] == 1, ! PrimeQ@ n, PrimeQ[n + 2]]] (* Michael De Vlieger, Aug 09 2017, after Artur Jasinski at A002997 *)

PARI

isA002997(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
isok(n) = isprime(n+2) && isA002997(n)

Extensions

More terms from Robert Israel, Jun 03 2019