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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). 8
3, 7, 13, 19, 37, 43, 79, 163, 223, 229, 608981812891, 608981812951, 608981812993, 608981813507, 608981813621, 608981813819, 608981813837, 608981813861, 608981813929, 608981813941, 608981814019, 608981814143, 608981814247, 608981814823 (list; graph; refs; listen; history; text; internal format)
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.]

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 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.

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 *)

CROSSREFS

Cf. A007352.

Sequence in context: A015913 A023200 A046136 * A252091 A217035 A134765

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

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Last modified December 16 07:41 EST 2017. Contains 296076 sequences.