|
| |
|
|
A130568
|
|
Beatty sequence 1+2*[n*phi], which contains infinitely many primes.
|
|
1
| |
|
|
1, 3, 7, 9, 13, 17, 19, 23, 25, 29, 33, 35, 39, 43, 45, 49, 51, 55, 59, 61, 65, 67, 71, 75, 77, 81, 85, 87, 91, 93, 97, 101, 103, 107, 111, 113, 117, 119, 123, 127, 129, 133, 135, 139, 143, 145, 149, 153, 155, 159, 161, 165, 169, 171, 175, 177, 181, 185, 187, 191, 195
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| The primes in this entirely odd sequence begin 3, 7, 13, 17, 19, 23, 29. By the theorems in Banks, there are an infinite number of primes in this sequence.
|
|
|
LINKS
| William D. Banks, Igor E. Shparlinski, Prime numbers with Beatty sequences, 7 Aug 2007.
|
|
|
FORMULA
| a(n) = 1+2*[n*phi] = 1+2*floor(n*phi), where phi = (1 + sqrt(5))/2.
|
|
|
EXAMPLE
| a(0) = 1 + 2*[0*phi] = 1 + 2*0 = 1.
a(1) = 1 + 2*[1*phi] = 1 + 2*[1.6180339887498948482045868343] = 1 + 2*1 = 3.
a(2) = 1 + 2*[2*phi] = 1 + 2*[3.2360679774997896964091736687] = 1 + 2*3 = 7.
a(3) = 1 + 2*[3*phi] = 1 + 2*[4.8541019662496845446137605030] = 1 + 2*4 = 9.
a(4) = 1 + 2*[4*phi] = 1 + 2*[6.4721359549995793928183473374] = 1 + 2*6 = 13.
a(5) = 1 + 2*[5*phi] = 1 + 2*[8.0901699437494742410229341718] = 1 + 2*8 = 17.
a(6) = 1 + 2*[6*phi] = 1 + 2*[9.7082039324993690892275210061] = 1 + 2*9 = 19.
a(7) = 1 + 2*[7*phi] = 1 + 2*[11.326237921249263937432107840] = 1 + 2*11 = 23.
a(8) = 1 + 2*[8*phi] = 1 + 2*[12.944271909999158785636694674] = 1 + 2*12 = 25.
a(9) = 1 + 2*[9*phi] = 1 + 2*[14.562305898749053633841281509] = 1 + 2*14 = 29.
a(10) = 1 + 2*[10*phi] = 1 + 2*[16.180339887498948482045868343] = 1 + 2*16 = 33.
|
|
|
MATHEMATICA
| Table[1 + 2*Floor[n*(Sqrt[5] + 1)/2], {n, 0, 80}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Aug 10 2007
|
|
|
CROSSREFS
| Cf. A001622.
Conjecture: Sequence gives n of A163873 whose connection to a(n) crosses (in the tree of A163873) another path. Is this generalizable in any way for A163874, A163875? [From Daniel Platt (d.platt(AT)web.de), Sep 14 2009]
Sequence in context: A140291 A032367 A063204 * A143803 A020497 A023490
Adjacent sequences: A130565 A130566 A130567 * A130569 A130570 A130571
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 09 2007
|
|
|
EXTENSIONS
| More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Aug 10 2007
|
| |
|
|
