A113755 Sequence implicit in Jaroma's corollary to Nagura's theorem on primes.

-1, -2, -4, -5, -9, -11, -14, -15, -18, -23, -24, -29, -31, -31, -32, -35, -37, -35, -36, -33, -27, -24, -17, -10, -2, 13, 34, 57, 88, 124, 157, 210, 273, 353, 441, 557, 693, 857, 1057, 1296

Offset

1,2

Comments

Tightening the bounds on J. Bertrand's 1845 conjecture that for any integer n > 3 there exists at least one prime between n and 2*n-2 (proved by P. Tchebechev in 1852), Nagura proved that, for n >= 25, there exists at least one prime number between n and (6/5)*n. John H. Jaroma gives an elementary proof by induction of the corollary: prime(n) < (1.2)^n for n > 25. Equivalently, this sequence, implicit in Nagura and Jaroma, is always positive after a(25). The minimum is reached with min((1.2)^n - prime(n)) = (1.2)^17 - 59 = -36.8138889.

Links

Table of n, a(n) for n=1..40.

John H. Jaroma, An Upper Bound on the n-th Prime, College Mathematics Journal 36.2 (2005) 158-159.

J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952) 177-181.

Formula

a(n) = floor((6/5)^n - prime(n)).

Example

a(1) = floor((1.2)^1 - prime(1)) = floor(1.2 - 2) = floor(-0.8) = -1.
a(2) = floor((1.2)^2 - prime(2)) = floor((1.2)^2 - 3) = floor(1.56) = -2.
a(3) = floor((1.2)^3 - prime(3)) = floor((1.2)^3 - 5) = floor(-3.27200) = -4.
a(4) = floor((1.2)^4 - prime(4)) = floor((1.2)^4 - 7) = floor(-4.9264) = -5.
a(25) = floor((1.2)^25 - prime(25)) = floor((1.2)^25 - 97) = floor(-1.60378336) = -2.
a(26) = floor((1.2)^26 - prime(26)) = floor((1.2)^26 - 101) = floor(13.47546) = +13.
a(40) = floor((1.2)^40 - prime(40)) = floor((1.2)^40 - 173) = floor(1296.77157) = 1296 = 6^4.

Mathematica

a[n_]:= Floor[(6/5)^n - Prime[n]]; Array[a, 40] (* Giovanni Resta, Jun 13 2016 *)

Crossrefs

Cf. A000040.

Sequence in context: A047378 A101155 A065825 * A124254 A192615 A258652

Adjacent sequences: A113752 A113753 A113754 * A113756 A113757 A113758

Keyword

easy,sign

Author

Jonathan Vos Post, Jan 16 2006

Extensions

a(6) corrected by Giovanni Resta, Jun 13 2016

Status

approved