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

1,1

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.

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

Example

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.

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]]

Prog

(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

Crossrefs

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

Sequence in context: A113864 A188754 A108310 * A330383 A146999 A147485

Adjacent sequences: A252395 A252396 A252397 * A252399 A252400 A252401

Keyword

nonn

Author

Jean-François Alcover, Dec 17 2014

Status

approved