A252398 Successive n with minimal relative distance |1-theta(n)/n|, where theta(n) = log(A034386(n)) is Chebyshev's theta function.

2, 3, 5, 7, 13, 19, 43, 47, 73, 103, 107, 109, 113, 199, 283, 467, 661, 887, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1621, 1627, 2803, 3931, 3947, 4273, 4289, 4297, 5867, 5869, 5881, 6373, 6379, 9439, 9473, 9479, 9497, 9551, 9859, 9931, 9949

Offset: 1

Author

Jean-François Alcover, Dec 17 2014

Keywords

• nonn

Comments

The first 10000 terms are the same as A108310 (see that sequence for comments). - Charles R Greathouse IV, Dec 18 2014

This sequence, unlike A108310, is presumably infinite; it is finite if and only if theta(n) = n for some number n.

Examples

			Given that 1 - theta(3)/3 = 1 - log(6)/3 = 0.40..., 1 - theta(4)/4 = 1 - log(6)/4 = 0.55... and 1 - theta(5)/5 = 1 - log(30)/5 = 0.31..., the next term after 3 is 5.
		

Links

• Jean-François Alcover and Charles R Greathouse IV, Table of n, a(n) for n = 1..5000 (first 88 terms from Alcover)
• Eric Weisstein's MathWorld, Chebyshev functions
• Wikipedia, Chebyshev function

Crossrefs

Cf. A016040, A035158, A057872, A083535, A108310, A215013.

Programs

Mathematica

(* Adapted from PARI *) Reap[For[record = 2; theta = 0; p = 2, p < 2 * 10^8, p = NextPrime[p], theta = theta + Log[p] //N; d = Abs[1 - theta/p]; If[d < record, record = d; Print[p]; Sow[p]]]][[2, 1]]

PARI

/* Note: This program may fail if you replace 1e6 with a number larger than 1e17, and will certainly fail at some point below 1e316. These large numbers are not remotely feasible at the moment. */
r=th=0; forprime(p=2,1e6, th+=log(p); t=th/p; if(t>r, r=t; print1(p", "); if(t>1, warning("theta(n) > n, possible missed terms")))) \\ Charles R Greathouse IV, Dec 17 2014