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

1,1

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)

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A007352.

Sequence in context: A015913 A023200 A046136 * A350591 A252091 A217035

Adjacent sequences: A098041 A098042 A098043 * A098045 A098046 A098047

Keyword

nonn

Author

Wayne VanWeerthuizen, Sep 10 2004

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

Status

approved