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A188465 Primes p such that p^2 divides 2^(2^(p-1)-1) - 1.

%I A188465 #40 Oct 27 2024 17:54:53
%S A188465 7,73,127,337,487,601,881,937,1801,2593,2647,3079,3943,4057,4201,6553,
%T A188465 7993,9199,10657,14407,15289,16759,18041,18121,20521,20809,21673,
%U A188465 22111,24967,25111,26407,28393,28729,36793,39367,41161,42463,47737,47881,49201,49297
%N A188465 Primes p such that p^2 divides 2^(2^(p-1)-1) - 1.
%C A188465 Primes p that divide 2^(2^(p-1)-1) - 1 are the same terms. Proof: p | 2^(2^(p-1)-1) - 1 iff ord_{p}(2) | 2^(p-1)-1, so p^2 | 2^(2^(p-1)-1) - 1, since p | 2^(p-1)-1 by FLT. - _Thomas Ordowski_, Sep 16 2024
%H A188465 Amiram Eldar, <a href="/A188465/b188465.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Arkadiusz Wesolowski)
%p A188465 isA188465 := proc(p) local m; if isprime(p) then m := modp(2 &^ ( (2 ^ (p-1))-1)-1,p) ; m := simplify(m) ; if m = 0 then true; else false; end if; else false; end if; end proc:
%p A188465 for i from 1 do p := ithprime(i) ; if isA188465(p) then printf("%d\n",p) ; end if; end do: # _R. J. Mathar_, Apr 10 2011
%t A188465 okQ[p_] := Module[{k = MultiplicativeOrder[2, p^2]}, PowerMod[2, p - 1, k] == 1]; Select[Prime[Range[5000]], okQ] (* _T. D. Noe_, Apr 11 2011 *)
%Y A188465 Cf. A001220.
%K A188465 nonn
%O A188465 1,1
%A A188465 _Arkadiusz Wesolowski_, Apr 10 2011