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A080076 Proth primes: primes of the form k*2^m + 1 with odd k < 2^m, m >= 1. 15

%I

%S 3,5,13,17,41,97,113,193,241,257,353,449,577,641,673,769,929,1153,

%T 1217,1409,1601,2113,2689,2753,3137,3329,3457,4481,4993,6529,7297,

%U 7681,7937,9473,9601,9857,10369,10753,11393,11777,12161,12289,13313

%N Proth primes: primes of the form k*2^m + 1 with odd k < 2^m, m >= 1.

%C Conjecture: a(n) ~ (n log n)^2 / 2. - _Thomas Ordowski_, Oct 19 2014

%H T. D. Noe, <a href="/A080076/b080076.txt">Table of n, a(n) for n = 1..10000</a>

%H C. Caldwell's The Top Twenty, <a href="http://primes.utm.edu/top20/page.php?id=66">Proth</a>.

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=fcVjitaM3LY">78557 and Proth Primes</a>, Numberphile video, 2017.

%H Max Lewis and Victor Scharaschkin, <a href="https://www.emis.de/journals/INTEGERS/papers/q80/q80.Abstract.html">k-Lehmer and k-Carmichael Numbers</a>, Integers, 16 (2016), #A80.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProthPrime.html">Proth Prime</a>

%p N:= 20000: # to get all terms <= N

%p S:= select(isprime, {seq(seq(k*2^m+1, k = 1 .. min(2^m, (N-1)/2^m), 2), m=1..ilog2(N-1))}):

%p sort(convert(S,list)); # _Robert Israel_, Feb 02 2016

%t r[p_, n_] := Reduce[p == (2*m + 1)*2^n + 1 && 2^n > 2*m + 1 && n > 0 && m >= 0, {a, m}, Integers]; r[p_] := Catch[ Do[ If[ r[p, n] =!= False, Throw[True]], {n, 1, Floor[Log[2, p]]}]]; A080076 = Reap[ Do[ p = Prime[k]; If[ r[p] === True, Sow[p]], {k, 1, 2000}]][[2, 1]] (* _Jean-Fran├žois Alcover_, Apr 06 2012 *)

%t nn = 13; Union[Flatten[Table[Select[1 + 2^n Range[1, 2^Min[n, nn - n + 1], 2], # < 2^(nn + 1) && PrimeQ[#] &], {n, nn}]]] (* _T. D. Noe_, Apr 06 2012 *)

%o (PARI) is_A080076(N)=isproth(N)&&isprime(N) \\ see A080075 for isproth(). - _M. F. Hasler_, Oct 18 2014

%Y Cf. A080075.

%Y Cf. A134876 (number of Proth primes), A214120, A239234.

%Y Cf. A248972.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Jan 24 2003

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Last modified November 20 23:06 EST 2019. Contains 329348 sequences. (Running on oeis4.)