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A337715 Numbers that are the product of two distinct odd numbers x*y such that 2^x (mod y) = 2^y (mod x) = 2.

%I A337715 #22 Sep 29 2024 11:10:38
%S A337715 341,525,651,765,1155,1387,1683,1935,2047,2701,3277,3751,4033,4165,
%T A337715 4305,4369,4455,4681,5461,5525,5715,6025,6643,7161,7239,7957,8265,
%U A337715 8321,8925,9471,9605,10261,10571,10965,12103,12325,13113,13747,13981,14491,15709,16275,16485
%N A337715 Numbers that are the product of two distinct odd numbers x*y such that 2^x (mod y) = 2^y (mod x) = 2.
%C A337715 Equivalently, terms of A176970 that are not square of odd primes. Every square of odd prime p^2 is a term of A176970, because by Fermat's little theorem, for p prime, then 2^p = 2 (mod p).
%C A337715 The super-Poulet numbers (A050217) form a subsequence with 341, 1387, 2047, 2701, ... (see example).
%e A337715 For 341 = 11 * 31 that is a super-Poulet:
%e A337715    2^11 (mod 31) = 2^31 (mod 11) = 2, hence 341 is a term;
%e A337715 For 525 = 3 * 5^2 * 7 = 15 * 35 = 21 * 25:
%e A337715    2^15 (mod 35) = 2^35 (mod 15) = 8, but
%e A337715    2^21 (mod 25) = 2^25 (mod 21) = 2, hence, 525 is a term.
%p A337715 test := proc(n) local d, q; if n::odd then for d in NumberTheory:-Divisors(n)
%p A337715 do q := iquo(n, d); if q > d and 2 &^ d mod q = 2 and 2 &^ q mod d = 2 then return true fi od fi; false end: select(test, [$(1..10000)]); # _Peter Luschny_, Sep 17 2020
%t A337715 okQ[x_, y_] := PowerMod[2, x, y] == PowerMod[2, y, x] == 2 && !PrimeQ[Sqrt[x*y]];
%t A337715 nn = 20000;
%t A337715 Union[Reap[Do[If[x*y < nn && okQ[x, y], Sow[x*y]], {x, 1, nn/3, 2}, {y, x, nn/3, 2}]][[2, 1]]] (* _Jean-François Alcover_, Sep 29 2024, after _Harvey P. Dale_ in A176970 *)
%o A337715 (PARI) isok(n) = {if ((n % 2), fordiv(n, d, if ((d > n/d) && (lift(Mod(2, d)^(n/d)) == 2) && (lift(Mod(2, n/d)^d) == 2), return(1));););} \\ _Michel Marcus_, Sep 17 2020
%Y A337715 Subsequence of A176970.
%Y A337715 A050217 (super-Poulet) is a subsequence.
%K A337715 nonn
%O A337715 1,1
%A A337715 _Bernard Schott_, Sep 16 2020
%E A337715 More terms from _Amiram Eldar_, Sep 16 2020