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A221743 Numbers k such that (6*k+1)*(12*k+1)*(18*k+1) is a Carmichael number which is the product of four prime numbers.

%I A221743 #28 Jul 09 2025 04:35:40
%S A221743 5,11,15,61,85,115,455,661,700,805,920,1225,1326,1910,2961,4935,5425,
%T A221743 6565,8175,10885,11375,12155,13230,18315,37800,39325,45325,59726,
%U A221743 69440,99645,113120,121365,129850,144685,211945,353465,378940,389896,392625
%N A221743 Numbers k such that (6*k+1)*(12*k+1)*(18*k+1) is a Carmichael number which is the product of four prime numbers.
%H A221743 Amiram Eldar, <a href="/A221743/b221743.txt">Table of n, a(n) for n = 1..500</a>
%H A221743 Umberto Cerruti, <a href="/A221743/a221743.pdf">Pseudoprimi di Fermat e numeri di Carmichael</a> (in Italian), 2013. The sequence is on page 11.
%H A221743 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%p A221743 with(numtheory);P:=proc(q)local a,b,k,ok,n;
%p A221743 for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
%p A221743 if issqrfree(a) and nops(b)=4 then ok:=1;
%p A221743 for k from 1 to 4 do if not type((a-1)/(b[k][1]-1),integer) then ok:=0;
%p A221743 break; fi; od; if ok=1 then print(n); fi;
%p A221743 fi; od; end: P(10^6); # _Paolo P. Lava_, Oct 11 2013
%t A221743 IsCarmichaelQ[n_] := Module[{f}, If[EvenQ[n] || PrimeQ[n], False, f = Transpose[FactorInteger[n]][[1]]; Union[Mod[n-1, f-1]] == {0}]]; n = 0; t = {}; While[Length[t] < 39, n++; c = (6*n + 1)*(12*n + 1)*(18*n + 1); If[SquareFreeQ[c] && Length[FactorInteger[c]] == 4 && IsCarmichaelQ[c], AppendTo[t, n]]]; t (* _T. D. Noe_, Jan 23 2013 *)
%o A221743 (Magma) [n: n in [1..4*10^5] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];
%Y A221743 Cf. A002997, A033502, A221742 (associated Carmichael numbers).
%Y A221743 Subsequence of A101187.
%K A221743 nonn
%O A221743 1,1
%A A221743 _Bruno Berselli_, Jan 23 2013, based on the Cerruti paper