A112714 Numbers of the form k*2^m-1 with k<2^m and k odd.

1, 3, 7, 11, 15, 23, 31, 39, 47, 55, 63, 79, 95, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255, 287, 319, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, 831, 863, 895, 927, 959, 991, 1023, 1087, 1151, 1215, 1279, 1343, 1407

Offset: 1

Author

Jose Brox (tautocrona(AT)terra.es), Dec 31 2005

Keywords

• easy
• nonn

Examples

			a(4)=7 because 7 = 1*2^3 - 1, with 1 < 2^3, and it is the fourth number of this form.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
• Eric Weisstein's World of Mathematics, Proth Numbers

Crossrefs

Cf. A080075.

Programs

Maple

N:= 2000: # to get all terms <= N
sort(convert({seq(seq(k*2^m-1,k=1..min((N+1)/2^m,2^m-1),2),m=1..ilog2(N+1))},list)); # Robert Israel, May 23 2017

Mathematica

Take[Sort@Flatten@Table[k*2^m - 1, {m, 0, 10}, {k, 1, 2^m - 1, 2}], 53] (* Robert G. Wilson v, Jan 02 2006 *)

PARI

for(n=2,8,for(k=2^(n-2)+1,2^n,print1(k*2^n-1","))) \\ Note that the first two terms (1,3) are not computed

Formula

a(n) = A080075(n)-2. - Thomas Ordowski, Aug 15 2025