A108310 -id:A108310 - cp's OEIS frontend

A215013 Primes p such that theta(p)/p < theta(q)/q for all primes q > p, where theta(x) = log(x#) is Chebyshev's theta function and x# is the product of primes <= x, A034386(x).

2, 3, 5, 11, 17, 29, 37, 41, 59, 67, 97, 101, 127, 149, 223, 227, 347, 349, 419, 431, 557, 563, 569, 587, 593, 599, 809, 821, 1423, 1427, 1429, 1433, 1447, 1451, 1481, 1973, 1987, 1993, 2657, 2659, 3299, 3301, 3307, 3449, 3457, 3461, 3511, 3527, 3529, 5381, 5387

Offset: 1

Charles R Greathouse IV, Jul 31 2012

Sequence is infinite. This follows from the Prime Number Theorem in the form theta(x) ~ x, together with (5.111) in Hardy & Littlewood (p. 184).

			theta(5)/5 = log(2*3*5)/5 = 0.680 is less than theta(7)/7 = log(2*3*5*7)/7 = 0.763..., theta(11)/11 = 0.704..., etc.
Using Dusart's explicit bounds it suffices to check up to 2569516503 to verify the first thousand terms of this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Pierre Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442 [math.NT], 2010.
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann Zeta-Function and the theory of the distribution of primes, Acta Mathematica 41 (1916), pp. 119-196.

Cf. A034386, A108310.

nMax = 1000; terms = 51; prd = 1; t = Table[p = Prime[n]; prd = prd*p; Log[prd]/p, {n, nMax}]; frst = 1; Table[mn = Min[Take[t, {frst, Length[t]}]]; pos = Position[t, mn, 1, 1][[1, 1]]; frst = pos + 1; Prime[pos], {terms}] (* T. D. Noe, Aug 02 2012 *)

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

Original entry on oeis.org

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

Jean-François Alcover, Dec 17 2014

nonn

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.

			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.

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

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

(* 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]]

/* 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

cp's OEIS Frontend

A215013 Primes p such that theta(p)/p < theta(q)/q for all primes q > p, where theta(x) = log(x#) is Chebyshev's theta function and x# is the product of primes <= x, A034386(x).

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