A176033 Numbers k such that 2^(2k-1) == 2 (mod 2k) and such that 2^(k-1) != 1 (mod k).

15, 85, 91, 435, 451, 703, 1247, 1271, 1581, 1695, 1891, 2071, 3133, 3367, 3683, 4795, 4859, 5551, 6643, 8695, 9061, 9131, 9211, 9605, 9919, 12403, 13019, 14351, 14701, 15051, 15211, 16021, 16471, 19669, 20191, 20485, 24727, 25351, 26335, 26599, 27511, 28645

Offset: 1

Author

Alzhekeyev Ascar M, Dec 06 2010

Keywords

• nonn

Comments

The associated value m = (2^(k-1) mod k) satisfy 1 < gcd(m-1, k) < k.

The selection criterion 2^(2k-1) == 2 (mod 2k) admits 3, 5, 7, 11, 13, 15, 17, etc.

Expect that the sequences will be infinite only if the criterion has the form 2^(2k-1) == 2^m (mod 2k) where m - an integer (1, 2, ...), otherwise the sequence will be limited. For example, for criterion 2^(2k-1) == 14 (mod 2k), the sequence begins 9, 27, 7281, 19143.

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Set difference of A020136 and A001567. - Max Alekseyev, Nov 06 2017

Programs

Maple

select(n -> 2 &^ (2*n-1) - 2 mod (2*n) = 0 and 2 &^ (n-1) -1 mod n <> 0, [seq(n,n=3..10^5,2)]); # Robert Israel, Nov 06 2017

Mathematica

Select[Range[30000],PowerMod[2,2#-1,2#]==2&&PowerMod[2,#-1,#]!=1&] (* Harvey P. Dale, Jul 06 2025 *)

PARI

alist(m) = {for (n=1, m, v = 2^(2*n-1); if ((v % (2*n) == 2), k = 2^(n-1) % n; if (k > 1, print1(n, ", "););););} \\ Michel Marcus, Jan 24 2013

Extensions

More terms from Michel Marcus, Jan 24 2013