A187849 Integers k such that 2^(k-1) == 1 (mod k) and 2^(m-1) == 1 (mod m), where m = k*(A000265(k-1) - 1) + 1 and A000265 gives the odd part of its argument.

563, 1291, 1733, 1907, 2477, 2609, 2693, 2837, 3533, 3677, 4157, 4517, 5693, 12809, 15077, 19997, 25603, 28517, 29573, 29837, 31517, 32237, 32717, 34949, 37277, 43613, 43973, 44453, 50333, 52253, 62477, 68213, 69197, 72893, 74717

Offset

1,1

Comments

The first condition of the definition means terms k are either primes or Fermat pseudoprimes to base 2 (see A001567). If the parameter m is also prime or Fermat pseudoprime to base 2, then k belongs to this sequence.

Composites in this sequence: 1771946607940820033, 14356915031659973281, ... - Max Alekseyev, Jun 04 2011

If the even part of k-1 is 2 and the odd part, j, is a composite number, then j == 0 (mod 3). [Is this heuristics or strict? - R. J. Mathar, Apr 04 2011] Terms to illustrate these cases:

k = 1291 = 2 * 645 + 1; j = 645 = 215 * 3.

k = 25603 = 2 * 12801 + 1; j = 12801 = 4267 * 3.

k = 424843 = 2 * 212421 + 1; j = 212421 = 70807 * 3.

k = 579883 = 2 * 289941 + 1; j = 289941 = 96647 * 3.

k = 4325443 = 2 * 2162721 + 1; j = 2162721 = 720907 * 3.

Links

Alzhekeyev Ascar M, Table of n, a(n) for n = 1..2297

Maple

isA187849 := proc(n) local redn, k, p, m; if modp(2^(n-1), n) = 1 then redn := n-1 ; k := A007814(redn) ; p := (n-1)/2^k ; m := n*(p-1)+1 ; is( modp(2^(m-1), m) = 1 ); else false; end if; end proc:
for n from 1 do if isA187849(n) then print(n); end if; end do: # R. J. Mathar, Mar 30 2011

Crossrefs

Cf. A000265, A001567, A007814.

Sequence in context: A232679 A135437 A142759 * A237029 A183732 A014361

Adjacent sequences: A187846 A187847 A187848 * A187850 A187851 A187852

Keyword

nonn

Author

Alzhekeyev Ascar M, Mar 14 2011

Extensions

Edited by Max Alekseyev, Jun 04 2011

Edited by Peter Munn, Aug 27 2025

Status

approved