A135437
Primes with a twin Carmichael number: primes p such that p-2 or p+2 are Carmichael numbers.
Original entry on oeis.org
563, 1103, 2467, 2819, 6599, 29339, 41039, 52631, 62743, 172079, 188459, 278543, 340559, 488879, 656599, 656603, 670031, 1033667, 1909003, 2100899, 3146219, 5048999, 6049679, 8719307, 10024559, 10402559, 10877579, 11119103, 12261059, 14913989, 15247619
Offset: 1
Pierre CAMI, Dec 14 2007, corrected Jun 22 2008; Sep 17 2008
563 is in the sequence since it is a prime number, and 563 - 2 = 561 is a Carmichael number.
1103 is in the sequence since it is a prime number, and 1103 + 2 = 1105 is a Carmichael number.
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s = {}; carmichaelQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Do[If[carmichaelQ[n], If[PrimeQ[n - 2], AppendTo[s, n - 2]]; If[PrimeQ[n + 2], AppendTo[s, n + 2]]], {n, 10^6}]; s (* Amiram Eldar, Jul 07 2019 *)
A290692
Carmichael numbers of the form p - 2 where p is a prime number.
Original entry on oeis.org
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
- 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.
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# 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
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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 *)
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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)
A308086
Carmichael numbers c such that c-4, c-2 and c+2 are primes.
Original entry on oeis.org
656601, 11512252145095521, 35151891169379601, 89283676825965441, 209606994019068801, 584047819872236721, 627126355430628801, 1107574117930742001, 1152431453119654401, 2990125943388676401, 6919232969930803761
Offset: 1
656601 = 3*11*101*197 is a term because 656597 and 656599 are twin primes, 656601 is a Carmichael number, and 656603 is also a prime.
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