A006992 Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.

2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 5003, 9973, 19937, 39869, 79699, 159389, 318751, 637499, 1274989, 2549951, 5099893, 10199767, 20399531, 40799041, 81598067, 163196129, 326392249, 652784471, 1305568919, 2611137817

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• nice

Comments

a(n) < a(n+1) by Bertrand's postulate (Chebyshev's theorem). - Jonathan Sondow, May 31 2014

Let b(n) = 2^n - a(n). Then b(n) >= 2^(n-1) - 1 and b(n) is a B_2 sequence: 0, 1, 3, 9, 19, 41, 85, 173, 349, ... - Thomas Ordowski, Sep 23 2014 See the link for B_2 sequence.

These primes can be obtained of exclusive form using a restricted variant of Rowland's prime-generating recurrence (A106108), making gcd(n, a(n-1)) = -1 when GCDs are greater than 1 and less than n (see program). These GCDs are also a divisor of each odd number from a(n) + 2 to 2*a(n-1) - 1 in reverse order, so that this subtraction with -1's invariably leads to the prime. - Manuel Valdivia, Jan 13 2015

First row of array in A229607. - Robert Israel, Mar 31 2015

Named after the French mathematician Joseph Bertrand (1822-1900). - Amiram Eldar, Jun 10 2021

References

• Martin Aigner and Günter M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 7.
• Martin Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), page 115. [From Martin Griffiths, Mar 28 2009]
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 344.
• Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 189.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 100 terms from T. D. Noe)
• Paul Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged, Vol. 5 (1932), pp. 194-198.
• Paul Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., Vol. 9 (1934), pp. 282-288.
• Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., Vol. 11 (1919), pp. 181-182.
• Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq., Vol. 15 (2012) Article 12.5.4.
• Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
• Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, Vol. 116, No. 7 (2009), pp. 630-635.
• Jonathan Sondow and Eric Weisstein, MathWorld: Bertrand's Postulate.
• Eric Weisstein's World of Mathematics, B2 Sequence.
• Robert G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993.

Crossrefs

Cf. A055496, A163961.

See A185231 for another version.

Cf. A007917, A229607, A295262.

Programs

Haskell

a006992 n = a006992_list !! (n-1)
a006992_list = iterate (a007917 . (* 2)) 2
-- Reinhard Zumkeller, Sep 17 2014

Maple

A006992 := proc(n) option remember; if n=1 then 2 else prevprime(2*A006992(n-1)); fi; end;

Mathematica

bertrandPrime[1] = 2; bertrandPrime[n_] := NextPrime[ 2*a[n - 1], -1]; Table[bertrandPrime[n], {n, 40}]
(* Second program: *)
NestList[NextPrime[2#, -1] &, 2, 40] (* Harvey P. Dale, May 21 2012 *)
k = 3; a[n_] := If[GCD[n,k] > 1 && GCD[n, k] < n, -1, GCD[n, k]]; Select[Differences@Table[k = a[n] + k, {n, 2611137817}], # > 1 &] (* Manuel Valdivia, Jan 13 2015 *)

PARI

print1(t=2);for(i=2,60,print1(", "t=precprime(2*t))) \\ Charles R Greathouse IV, Apr 01 2013

Python

from sympy import prevprime
l = [2]
for i in range(1, 51):
    l.append(prevprime(2 * l[i - 1]))
print(l) # Indranil Ghosh, Apr 26 2017

Formula

a(n+1) = A007917(2*a(n)). - Reinhard Zumkeller, Sep 17 2014

Limit_{n -> infinity} a(n)/2^n = 0.303976447924... - Thomas Ordowski, Apr 05 2015

Extensions

Definition completed by Jonathan Sondow, May 31 2014

B_2 sequence link added by Wolfdieter Lang, Oct 09 2014