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

Author

Charles R Greathouse IV, Jul 31 2012

Keywords

• nonn
• nice

Comments

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).

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A034386, A108310.

Programs

Mathematica

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 *)