A098044 Odd primes p such that Pi_{3,1}(p) = Pi_{3,2}(p) - 1, where Pi_{m,n}(p) denotes the number of primes q <= p with q == n (mod m).

3, 7, 13, 19, 37, 43, 79, 163, 223, 229, 608981812891, 608981812951, 608981812993, 608981813507, 608981813621, 608981813819, 608981813837, 608981813861, 608981813929, 608981813941, 608981814019, 608981814143, 608981814247, 608981814823

Offset: 1

Author

Wayne VanWeerthuizen, Sep 10 2004

Keywords

• nonn

Comments

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, p. 275.]

From M. F. Hasler, May 10 2021: (Start)

Using the data a(1..85509) computed by D. Johnson, the major gaps are as follows:

between and size of the gap

a(10) = 229 a(11) = 608981812891 609e9

a(11491) = 609340824721 a(11492) = 610704087667 1.3e9

a(21325) = 610968213803 a(21326) = 6148171711471 5.5e12

a(38653) = 6148988909519 a(38654) = 6149773241849 784e6

a(49417) = 6151116531611 a(49418) = 6151907045033 791e6

a(65479) = 6152794922413 a(65480) = 6153794890993 1.0e9

a(73171) = 6154352395729 a(73172) = 6155140151519 788e6

a(85509) = 6156051951809 ??? ???. (End)

Examples

			There are five odd primes <= 37 of the form 3n+1. They are 7, 13, 19, 31, 37. There are five odd 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.
		

References

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

Links

• Donovan Johnson, Table of n, a(n) for n = 1..85509

Crossrefs

Cf. A007352.

Programs

Mathematica

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 (* Robert G. Wilson v, Sep 11 2004 *)

PARI

N=100; c=1; forprime(p=3,, if(p%3>1,c++,c--)||print1(p",")||N--||break) \\ Takes only ~1 second up to 1e8, but to see the next terms, beyond 6e11, replace p=3 with p=608981812891. - M. F. Hasler, May 10 2021

Formula

For n>1, a(n) = A000040(A096629(n-1)).

Extensions

Edited and extended by Robert G. Wilson v, Sep 11 2004

Initial entry 3 added by David Wasserman, Nov 07 2007

Edited and terms a(11) onward added by Max Alekseyev, Feb 09 2011