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

341, 525, 651, 765, 1155, 1387, 1683, 1935, 2047, 2701, 3277, 3751, 4033, 4165, 4305, 4369, 4455, 4681, 5461, 5525, 5715, 6025, 6643, 7161, 7239, 7957, 8265, 8321, 8925, 9471, 9605, 10261, 10571, 10965, 12103, 12325, 13113, 13747, 13981, 14491, 15709, 16275, 16485

Offset: 1

Author

Bernard Schott, Sep 16 2020

Keywords

• nonn

Comments

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).

The super-Poulet numbers (A050217) form a subsequence with 341, 1387, 2047, 2701, ... (see example).

Examples

			For 341 = 11 * 31 that is a super-Poulet:
   2^11 (mod 31) = 2^31 (mod 11) = 2, hence 341 is a term;
For 525 = 3 * 5^2 * 7 = 15 * 35 = 21 * 25:
   2^15 (mod 35) = 2^35 (mod 15) = 8, but
   2^21 (mod 25) = 2^25 (mod 21) = 2, hence, 525 is a term.
		

Crossrefs

Subsequence of A176970.

A050217 (super-Poulet) is a subsequence.

Programs

Maple

test := proc(n) local d, q; if n::odd then for d in NumberTheory:-Divisors(n)
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

Mathematica

okQ[x_, y_] := PowerMod[2, x, y] == PowerMod[2, y, x] == 2 && !PrimeQ[Sqrt[x*y]];
nn = 20000;
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 *)

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

Extensions

More terms from Amiram Eldar, Sep 16 2020