A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

1, 1, 3, 3, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, 225, 227, 237, 231, 245, 251, 257, 267, 265, 273, 279

Offset

1,3

Comments

The theorem that a(n) > 0 for all n is known as "Bertrand's Postulate", and was proved by Tchebycheff in 1852.

The analog for Ramanujan primes is Paksoy's theorem that 2*R(n) - R(n+1) > 0 for n > 1. See A233822. - Jonathan Sondow, Dec 16 2013

References

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939.

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

Formula

a(n) = A000040(n) - A001223(n). - Zak Seidov, Sep 07 2012

a(n) = 2*A000040(n) - A000040(n+1). - Zak Seidov, May 12 2020

a(n) = A098764(n) - A000040(n). - Anthony S. Wright, Feb 19 2024

Maple

a:= n-> (p-> 2*p(n)-p(n+1))(ithprime):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2022

Prog

(PARI) a(n) = 2*prime(n) - prime(n + 1); \\ Harry J. Smith, Aug 03 2009
(Haskell)
a062234 n = a062234_list !! (n-1)
a062234_list = zipWith (-) (map (* 2) a000040_list) (tail a000040_list)
-- Reinhard Zumkeller, May 31 2015

Crossrefs

Cf. A000040, A001223, A215808 (prime terms), A233822.

Cf. A098764, A100484, A258383 (run lengths), A258432, A258469, A257762, A258449, A257892, A257951.

Sequence in context: A183429 A107443 A204099 * A168329 A161828 A219128

Adjacent sequences: A062231 A062232 A062233 * A062235 A062236 A062237

Keyword

easy,nonn

Author

Reinhard Zumkeller, Jun 29 2001

Extensions

Edited by N. J. A. Sloane, Feb 24 2023

Status

approved