A006992 - OEIS

A006992

Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.
(Formerly M0675)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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

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 *)

PROG

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

(Haskell)

a006992 n = a006992_list !! (n-1)

a006992_list = iterate (a007917 . (* 2)) 2

-- Reinhard Zumkeller, Sep 17 2014

(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