A290692 - OEIS

%I #42 Apr 22 2024 13:48:18

%S 561,2465,656601,1909001,174352641,230996949,275283401,939947009,

%T 1534274841,3264820001,5860426881,6025532241,25536531021,36709177121,

%U 53388707681,54519328481,56222911361,101536702401,105528976961,180481509681,196866607601,239862350001,329245587161,347469383801,347511324161

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

%C 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.

%C 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).

%H Amiram Eldar, <a href="/A290692/b290692.txt">Table of n, a(n) for n = 1..5901</a> (terms below 10^22 calculated using data from Claude Goutier; terms 1..591 from Robert Israel)

%H Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.

%H R. G. E. Pinch, Carmichael numbers up to <a href="http://s369624816.websitehome.co.uk/rgep/carmichael-16.gz">10^16</a>, <a href="http://s369624816.websitehome.co.uk/rgep/carmichael17.gz">10^16 to 10^17</a>, <a href="http://s369624816.websitehome.co.uk/rgep/carmichael18.gz">10^17 to 10^18</a>

%H Andrzej Rotkiewicz, <a href="http://dml.cz/dmlcz/137472">On pseudoprimes having special forms and a solution of K. Szymiczek's problem</a>, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.

%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.

%p # Using data file from Richard Pinch

%p infile:= "carmichael-16": Res:= NULL;

%p do

%p   S:= readline(infile);

%p   if S = 0 then break fi;

%p   L:= sscanf(S,"%d");

%p   if nops(L) <> 1 then break fi;

%p   if isprime(L[1]+2) then Res:= Res, L[1]; fi

%p od:

%p Res; # _Robert Israel_, Jun 03 2019

%t 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 *)

%o (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}

%o isok(n) = isprime(n+2) && isA002997(n)

%Y Cf. A002997, A272754, A287591.

%K nonn

%O 1,1

%A _Altug Alkan_, Aug 09 2017

%E More terms from _Robert Israel_, Jun 03 2019