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A328665 Least super-Poulet number (A050217) with n distinct prime factors.

%I A328665 #17 Feb 16 2025 08:33:58
%S A328665 341,294409,9972894583,1264022137981459,14054662152215842621
%N A328665 Least super-Poulet number (A050217) with n distinct prime factors.
%C A328665 a(7) <= 1842158622953082708177091, and a(8) <= 317565023788749598474704753433331761 (from Michon's site).
%C A328665 From _Daniel Suteu_, Oct 28 2019: (Start)
%C A328665 a(8)  <= 192463418472849397730107809253922101,
%C A328665 a(9)  <= 1347320741392600160214289343906212762456021,
%C A328665 a(10) <= 70865138168006643427403953978871929070133095474701,
%C A328665 a(11) <= 3363391752747838578311772729701478698952546288306688208857,
%C A328665 a(12) <= 132153369641266990823936945628293401491197666138621036175881960329,
%C A328665 a(13) <= 9105096650335639994239038954861714246150666715328403635257215036295306537. (End)
%H A328665 Gérard P. Michon, <a href="http://www.numericana.com/answer/pseudo.htm#super">Super-pseudoprimes to Base a</a>, Numericana, 2005.
%H A328665 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Super-PouletNumber.html">Super-Poulet Numbers</a>
%H A328665 Wikipedia, <a href="http://en.wikipedia.org/wiki/Super-Poulet_number">Super-Poulet number</a>
%t A328665 a[n_] := Module[{k=1}, While[PrimeNu[k] < n || PowerMod[2, k - 1, k] != 1 || Union @ PowerMod[2, Rest[Divisors[k]], k] != {2}, k++]; k]; Array[a, 3, 2]
%o A328665 (PARI) isok(k, n) = if (omega(k) == n, fordiv(k, d, if(Mod(2, d)^d!=2, return(0))); return(1));
%o A328665 a(n) = my(k=4); while (!isok(k, n), k++); k; \\ _Michel Marcus_, Oct 28 2019
%o A328665 (PARI) isupperbound(n,k) = my(f=factor(k)); omega(f) == n && Mod(2, k)^gcd(vector(#f~, i, f[i,1]-1)) == 1; \\ _Daniel Suteu_, Oct 28 2019
%Y A328665 Cf. A001567, A007011, A050217, A178997, A300327.
%K A328665 nonn,more
%O A328665 2,1
%A A328665 _Amiram Eldar_, Oct 24 2019