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A080075 Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k. 12
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014

A Proth number is a square iff it is of the form (2^(m-1)+-1)*2^(m+1)+1 = 4^m+-2^(m+1)+1 = (2^m+-1)^2 for m > 1. See A086341. - Thomas Ordowski, Apr 22 2019

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).

Eric Weisstein's World of Mathematics, Proth Number

Wikipedia, Proth number

FORMULA

a(n) = A116882(n+1)+1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010

a(n) = A157892(n)*2^A157893(n) + 1. - M. F. Hasler, Aug 16 2010

MATHEMATICA

Select[Range[3, 1500, 2], And[OddQ[#[[1]] ], #[[-1]] >= 1, 2^#[[-1]] > #[[1]] ] &@ Append[QuotientRemainder[#1, 2^#2], #2] & @@ {#, IntegerExponent[#, 2]} &[# - 1] &] (* Michael De Vlieger, Nov 04 2019 *)

PROG

(PARI) isproth(x)={ (x>1) && !bittest(x--, 0) && (x>>valuation(x, 2))^2 < x } \\ M. F. Hasler, Aug 16 2010; edited by Michel Marcus, Apr 23 2019

CROSSREFS

Cf. A080076, A112714.

Sequence in context: A211340 A061571 A049690 * A007664 A215812 A114395

Adjacent sequences:  A080072 A080073 A080074 * A080076 A080077 A080078

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Jan 24 2003

STATUS

approved

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Last modified December 1 03:31 EST 2020. Contains 338833 sequences. (Running on oeis4.)