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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
(list;
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refs;
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history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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Named after the French farmer and self-taught mathematician François Proth (1852-1879). - Amiram Eldar, Jun 05 2021
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Chris K. Caldwell's The Top Twenty, Proth.
Bertalan Borsos, Attila Kovács and Norbert Tihanyi, Tight upper and lower bounds for the reciprocal sum of Proth primes, The Ramanujan Journal (2022).
James Grime and Brady Haran, 78557 and Proth Primes, Numberphile video, 2017.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Max Lewis and Victor Scharaschkin, k-Lehmer and k-Carmichael Numbers, Integers, Vol. 16 (2016), #A80.
Rogério Paludo and Leonel Sousa, Number Theoretic Transform Architecture suitable to Lattice-based Fully-Homomorphic Encryption, 2021 IEEE 32nd Int'l Conf. Appl.-specific Sys., Architectures and Processors (ASAP) 163-170.
François Proth, Théorèmes sur les nombres premiers, Comptes rendus de l'Académie des Sciences de Paris, Vol. 87 (1878), p. 926.
Tsz-Wo Sze, Deterministic Primality Proving on Proth Numbers, arXiv:0812.2596 [math.NT], 2009.
Eric Weisstein's World of Mathematics, Proth Prime.
Wikipedia, Proth prime.
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FORMULA
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Conjecture: a(n) ~ (n log n)^2 / 2. - Thomas Ordowski, Oct 19 2014
Sum_{n>=1} 1/a(n) is in the interval (0.7473924793, 0.7473924795) (Borsos et al., 2022). - Amiram Eldar, Jan 29 2022
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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
next_A080076(N)={until(isprime(N=next_A080075(N)), ); N}
A080076_first(N)=vector(N, i, N=if(i>1, next_A080076(N), 3)) \\ M. F. Hasler, Jul 07 2022, following a suggestion from Bill McEachen
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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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