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

Author

Reinhard Zumkeller, Jun 29 2001

Keywords

• easy
• nonn

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

• Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Harry J. Smith)

Crossrefs

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

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

When negated, forms the left edge of irregular triangle A252750, and also the leftmost column of square array A372562.

Programs

Haskell

a062234 n = a062234_list !! (n-1)
a062234_list = zipWith (-) (map (* 2) a000040_list) (tail a000040_list)
-- Reinhard Zumkeller, May 31 2015

Maple

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

Mathematica

Table[2*Prime[n]-Prime[n+1],{n,60}] (* James C. McMahon, Apr 27 2024 *)
2#[[1]]-#[[2]]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Jul 29 2024 *)
ListConvolve[{-1, 2}, Prime[Range[100]]] (* Paolo Xausa, Nov 02 2024 *)

PARI

a(n) = 2*prime(n) - prime(n + 1); \\ Harry J. Smith, Aug 03 2009

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

Extensions

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