OFFSET
1,1
COMMENTS
Named after the French farmer and self-taught mathematician François Proth (1852-1879). - Amiram Eldar, Jun 05 2021
LINKS
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.
Youngik Lee, Hyperbolic Primality Test and Catalan-Mersenne Number Conjecture, Brown Univ., Preprints.org (2024). See p. 6.
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.
Hermann Stamm-Wilbrandt, a080076.json, sorted array of all 122,742 Proth primes less than 2^40 (> 10^12).
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.
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) is_A080076(N)=isproth(N)&&isprime(N) \\ see A080075 for isproth(). - M. F. Hasler, Oct 18 2014
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 24 2003
STATUS
approved