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

1,1

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A281189 A206383 A020136 * A067401 A206169 A160599

Adjacent sequences: A176030 A176031 A176032 * A176034 A176035 A176036

Keyword

nonn

Author

Alzhekeyev Ascar M, Dec 06 2010

Extensions

More terms from Michel Marcus, Jan 24 2013

Status

approved