1,2

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.

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.

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

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.

Cf. A000040.

Sequence in context: A036795 A024618 A089221 this_sequence A091271 A063985 A050052

Adjacent sequences: A113752 A113753 A113754 this_sequence A113756 A113757 A113758

easy,sign

Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 16 2006