A287591
Carmichael numbers k such that k-2 and k+2 are both primes.
Original entry on oeis.org
656601, 25536531021, 8829751133841, 60561233400921, 79934093254401, 352609909731201, 598438077923841, 976515437206401, 2122162714918401, 2789066007968241, 3767175573114801, 7881891474971361, 10740122274670881, 11512252145095521, 16924806963384321
Offset: 1
656601 is in the sequence since it is a Carmichael number (A002997) and both 656599 and 656603 are primes.
- Amiram Eldar, Table of n, a(n) for n = 1..282 (terms below 10^22, calculated using data from Claude Goutier)
- Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
- R. G. E. Pinch, Tables relating to Carmichael numbers.
- 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.
A194231
Numbers k such that at least one of k and k+2 is composite, while for every b coprime to k*(k+2), b^(k-1) == 1 (mod k) and b^(k+1) == 1 (mod k+2).
Original entry on oeis.org
561, 1103, 2465, 2819, 6599, 29339, 41039, 52631, 62743, 172079, 188459, 278543, 340559, 488879, 656599, 656601, 670031, 1033667, 1909001, 2100899, 3146219, 5048999, 6049679, 8719307, 10024559, 10402559, 10877579, 11119103, 12261059, 14913989, 15247619
Offset: 1
-
with(numtheory):
ic:= proc(n) local p;
if not issqrfree(n) then false
else for p in factorset(n) do
if irem (n-1, p-1)<>0 then return false fi
od; true
fi
end:
a:= proc(n) option remember; local k;
for k from 2 +`if`(n=1, 1, a(n-1)) by 2 while
isprime(k) and isprime(k+2) or not (ic(k) and ic(k+2))
do od; k
end:
seq(a(n), n=1..10); # Alois P. Heinz, Oct 12 2011
-
terms = 31; bMax = 20(* sufficient for 31 terms *); coprimes[n_] := Select[ Range[bMax], CoprimeQ[#, n]&]; Reap[For[n = m = 1, m <= terms, n += 2, If[CompositeQ[n] || CompositeQ[n+2], If[AllTrue[coprimes[n(n+2)], PowerMod[#, n-1, n] == 1 && PowerMod[#, n+1, n+2] == 1&], Print["a(", m, ") = ", n]; Sow[n]; m++]]]][[2, 1]] (* Jean-François Alcover, Mar 28 2017 *)
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.
-
# 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
-
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 *)
-
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)
Showing 1-3 of 3 results.
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