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A080076
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Proth primes: primes of the form k*2^m + 1 with odd k < 2^m, m >= 1.
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18
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3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, 10369, 10753, 11393, 11777, 12161, 12289, 13313
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) ~ (n log n)^2 / 2. - Thomas Ordowski, Oct 19 2014
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
C. Caldwell's The Top Twenty, Proth.
James Grime and Brady Haran, 78557 and Proth Primes, Numberphile video, 2017.
Max Lewis and Victor Scharaschkin, k-Lehmer and k-Carmichael Numbers, Integers, 16 (2016), #A80.
Tsz-Wo Sze, Deterministic Primality Proving on Proth Numbers, arXiv:0812.2596 [math.NT], 2009.
Eric Weisstein's World of Mathematics, Proth Prime
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MAPLE
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N:= 20000: # to get all terms <= N
S:= select(isprime, {seq(seq(k*2^m+1, k = 1 .. min(2^m, (N-1)/2^m), 2), m=1..ilog2(N-1))}):
sort(convert(S, list)); # Robert Israel, Feb 02 2016
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MATHEMATICA
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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-François Alcover, Apr 06 2012 *)
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 *)
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PROG
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(PARI) is_A080076(N)=isproth(N)&&isprime(N) \\ see A080075 for isproth(). - M. F. Hasler, Oct 18 2014
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CROSSREFS
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Cf. A080075.
Cf. A134876 (number of Proth primes), A214120, A239234.
Cf. A248972.
Sequence in context: A180008 A089996 A307512 * A128339 A147506 A282960
Adjacent sequences: A080073 A080074 A080075 * A080077 A080078 A080079
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein, Jan 24 2003
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STATUS
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approved
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