1,1

This is the break-even point among the odd primes of the form 3n+1 versus primes the form 3n+2.

"On the average Pi_{3,2}(x) - Pi_{3,1}(x) is asymptotically sqrt(x)/Log(x). However, Hudson (with the help of Schinzel) showed in 1985 that lim_{x --> inf} (Pi_{3,2}(x) - Pi_{3,1}(x))/ sqrt(x)/Log(x) does not exist (in particular, it is not equal to 1)." Ribenboim, pg 275.

Up to p_10^7, Pi_{3,1} = 4999504 and Pi_{3,2} = 5000494. Recall that 2 is an even prime and 3 = 0 (mod 3).

Pi_{3,1} (10^10) = 227523123 & Pi_{3,2} (10^10) = 227529387 and Pi(10^10) = 455052511.

P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1995, page 274.

There are five primes <= 37 of the form 3n+1. They are 7, 13, 19, 31, 37. There are five primes <= 37 of the form 3n+2. They are 5, 11, 17, 23, 29. Therefore 37 is a "break-even" point among the odd primes.

p31 = p32 = 0; lst = {}; Do[p = Prime[n]; Switch[ Mod[p, 3], 1, p31++, 2, p32++ ]; If[ p31==p32, AppendTo[lst, p]], {n, 3, 10^7}]; lst (from Robert G. Wilson v Sep 11 2004)

Cf. prime races: A007352.

Sequence in context: A015913 A023200 A046136 this_sequence A134765 A171817 A023230

Adjacent sequences: A098041 A098042 A098043 this_sequence A098045 A098046 A098047

nonn

Wayne VanWeerthuizen (waynemv(AT)yahoo.com), Sep 10 2004

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 11 2004

Initial entry 3 added by David Wasserman (dwasserm(AT)earthlink.net), Nov 07 2007