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A006992 Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.
(Formerly M0675)
32
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

REFERENCES

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 7.

M. 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.

I. Niven and H. 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

T. D. Noe and Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 100 terms from T. D. Noe)

P. Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged 5 (1932), pp. 194-198.

P. Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., 9 (1934), 282-288.

Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.

Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 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, 116 (2009), 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

Lim_{n -> inf} 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]

i=1

while i<=50:

    l+=[prevprime(2*l[i - 1]), ]

    i+=1

print l # Indranil Ghosh, Apr 26 2017

CROSSREFS

Cf. A055496, A163961.

See A185231 for another version.

Cf. A007917, A229607, A295262.

Sequence in context: A126092 A132394 A295262 * A185231 A080190 A076994

Adjacent sequences:  A006989 A006990 A006991 * A006993 A006994 A006995

KEYWORD

nonn,nice

AUTHOR

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

EXTENSIONS

Definition completed by Jonathan Sondow, May 31 2014

B_2 sequence link added by Wolfdieter Lang, Oct 09 2014

STATUS

approved

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Last modified June 25 05:41 EDT 2019. Contains 324346 sequences. (Running on oeis4.)