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%I #73 Apr 24 2024 08:19:19
%S 3,5,9,13,17,25,33,41,49,57,65,81,97,113,129,145,161,177,193,209,225,
%T 241,257,289,321,353,385,417,449,481,513,545,577,609,641,673,705,737,
%U 769,801,833,865,897,929,961,993,1025,1089,1153,1217,1281,1345,1409
%N Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.
%C 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
%H Charles R Greathouse IV, <a href="/A080075/b080075.txt">Table of n, a(n) for n = 1..10000</a>
%H Bertalan Borsos, Attila Kovács and Norbert Tihanyi, <a href="https://doi.org/10.1007/s11139-021-00536-2">Tight upper and lower bounds for the reciprocal sum of Proth primes</a>, The Ramanujan Journal (2022).
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 38.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3339313">Discussion of the groupoid of Proth numbers (OEIS A080075)</a>, Politecnico di Torino, Italy (2019).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProthNumber.html">Proth Number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Proth_number">Proth number</a>.
%F a(n) = A116882(n+1)+1. - _Klaus Brockhaus_, _Georgi Guninski_ and _M. F. Hasler_, Aug 16 2010
%F a(n) = A157892(n)*2^A157893(n) + 1. - _M. F. Hasler_, Aug 16 2010
%F a(n) ~ n^2/2. - _Thomas Ordowski_, Oct 19 2014
%F Sum_{n>=1} 1/a(n) = 1.09332245643583252894473574405304699874426408312553... (Borsos et al., 2022). - _Amiram Eldar_, Jan 29 2022
%F 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
%t 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 *)
%o (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
%o (PARI) next_A080075(N)=N+2^(exponent(N)\2+1)
%o A080075_first(N)=vector(N,i,if(i>1,next_A080075(N),3)) \\ _M. F. Hasler_, Jul 07 2022
%Y Cf. A080076, A086341, A112714, A116882, A157892, A157893.
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Jan 24 2003
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