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A221742 Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1) which are the product of four prime numbers.

%I A221742 #37 Jul 09 2025 04:35:33
%S A221742 172081,1773289,4463641,295643089,798770161,1976295241,122160500281,
%T A221742 374464040689,444722065201,676328168881,1009514855521,2382986541601,
%U A221742 3022286597929,9031805532361,33648448111489,155773422536761,206932492972801,366715617643441,708083570971801
%N A221742 Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1) which are the product of four prime numbers.
%H A221742 Amiram Eldar, <a href="/A221742/b221742.txt">Table of n, a(n) for n = 1..500</a> (terms 1..87 from Vincenzo Librandi)
%H A221742 Umberto Cerruti, <a href="/A221742/a221742.pdf">Pseudoprimi di Fermat e numeri di Carmichael</a> (in Italian), 2013. The sequence is on page 11.
%H A221742 <a href="https://oeis.org/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%p A221742 with(numtheory);P:=proc(q)local a,b,k,ok,n;
%p A221742 for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
%p A221742 if issqrfree(a) and nops(b)=4 then ok:=1;
%p A221742 for k from 1 to 4 do if not type((a-1)/(b[k][1]-1),integer) then ok:=0;
%p A221742 break; fi; od; if ok=1 then print(a); fi;
%p A221742 fi; od; end: P(10^6); # _Paolo P. Lava_, Oct 11 2013
%t A221742 g[n_] := (6*n+1)*(12*n+1)*(18*n+1); testQ[n_] := Block[{p,e}, {p, e} = Transpose@ FactorInteger@ n; e == {1,1,1,1} && Max[Mod[n-1, p-1]] == 0]; Select[g /@ Range[10^4], testQ] (* _Giovanni Resta_, May 21 2013 *)
%o A221742 (Magma) [c: n in [1..10^4] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];
%Y A221742 Cf. A002997, A033502, A221743 (associated k).
%Y A221742 Subsequence of A182087.
%K A221742 nonn
%O A221742 1,1
%A A221742 _Bruno Berselli_, Jan 23 2013, based on the Cerruti paper