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A098044
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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).
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8
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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; internal format)
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OFFSET
| 1,1
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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, pg 275.
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REFERENCES
| P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1995, page 274.
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LINKS
| Donovan Johnson, Table of n, a(n) for n = 1..85509
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FORMULA
| For n>1, a(n) = A000040(A096629(n-1)).
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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.
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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 (from Robert G. Wilson v Sep 11 2004)
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CROSSREFS
| Cf. A007352.
Sequence in context: A015913 A023200 A046136 * A134765 A023230 A194581
Adjacent sequences: A098041 A098042 A098043 * A098045 A098046 A098047
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KEYWORD
| nonn
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AUTHOR
| Wayne VanWeerthuizen (waynemv(AT)yahoo.com), Sep 10 2004
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EXTENSIONS
| 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
Edited and terms a(11) onward added by Max Alekseyev (maxale(AT)gmail.com), Feb 09 2011
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