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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 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

Bertalan Borsos, Attila Kovács and Norbert Tihanyi, Tight upper and lower bounds for the reciprocal sum of Proth primes, The Ramanujan Journal (2022).

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

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

Sum_{n>=1} 1/a(n) = 1.09332245643583252894473574405304699874426408312553... (Borsos et al., 2022). - Amiram Eldar, Jan 29 2022

a(n+1) = a(n) + 2^round(L(n)/2), where L(n) is the number of binary digits of a(n); equivalently, floor(log_2(a(n))/2 + 1) in the exponent. [Lemma 2.2 in Borsos et al.] - M. F. Hasler, Jul 07 2022

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) is_A080075 = isproth(x)={!bittest(x--, 0) && (x>>valuation(x+!x, 2))^2 < x } \\ M. F. Hasler, Aug 16 2010; edited by Michel Marcus, Apr 23 2019, M. F. Hasler, Jul 07 2022

next_A080075(N)=N+2^(exponent(N)\2+1)

A080075_first(N)=vector(N, i, if(i>1, next_A080075(N), 3)) \\ M. F. Hasler, Jul 07 2022

CROSSREFS

Cf. A080076, A086341, A112714, A116882, A157892, A157893.

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 October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)