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

1,1

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.

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 *)

Prog

(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)

Crossrefs

Cf. A002997, A272754, A287591.

Sequence in context: A097130 A110889 A205947 * A293622 A322130 A354609

Adjacent sequences: A290689 A290690 A290691 * A290693 A290694 A290695

Keyword

nonn,changed

Author

Altug Alkan, Aug 09 2017

Extensions

More terms from Robert Israel, Jun 03 2019

Status

approved