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Search: id:A164917
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| A164917 |
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Smallest number of steps to reach prime(n) by applying the map x->A060308(x) starting from any member of A164368. |
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6
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| 0, 1, 2, 3, 0, 4, 0, 1, 5, 0, 1, 2, 0, 6, 0, 1, 0, 2, 0, 0, 3, 1, 7, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 8, 0, 2, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 3, 9, 1, 3, 0, 0, 1, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 2, 0, 0, 0, 3, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 3, 0, 1, 2, 3, 1, 1, 0, 0, 2
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Starting from some prime, iterated application of A060308 (or of the equivalent A059788)
generates a chain of increasing prime numbers. The nature of these chains is
to reach higher in the list of primes, sometimes "over-fulfilling" Betrand's
postulate by skipping some nearer primes, almost doubling of possible. On the other hand, A164368
contains the primes that would be skipped by a chain which contains the prime slightly
above half of their value. The sequence shows how far up in chains starting
from some member of A164368 we find prime(n), or equivalently, how many inverse
applications of the map we need to hit a member of A164368 if starting at prime(n).
Note that by construction A164368(k) starts with the smallest prime that is not member
of any chain started from any previous A164368. So each prime exists at some place
in one of these chains, and the number of steps a(n) to reach it from the start of its chain is well defined.
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LINKS
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V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319
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EXAMPLE
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The first prime chains of the mapping with A060308 initialized with members of A164368 are
2->3->5->7->13->23->43->83->163->317->631->1259->2503->..
11->19->37->73->139->277->547->1093->2179->4357->8713->17419->..
17->31->61->113->223->443->883->1759->3517->7027->14051->28099->..
29->53->103->199->397->787->1571->3137->6271->12541->25073->..
41->79->157->313->619->1237->2473->4943->9883->19763->39521->..
47->89->173->337->673->1327->2647->5281->10559->21107->..
The a(1) to a(4) representing the first 4 primes are all on the first chain, and
need 0 to 3 steps to be reached from 2 = A164368(1). a(5) asks for the number of steps
for A000040(5)=11 which is on the second chain, and needs 0 steps.
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MAPLE
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A060308 := proc(n) prevprime(2*n+1) ; end:
isA164368 := proc(p) local q ; q := nextprime(floor(p/2)) ; RETURN(numtheory[pi](2*q) -numtheory[pi](p) >= 1); end:
A164368 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1)) ; while not isA164368(a) do a := nextprime(a) ; od: RETURN(a) ; fi; end:
A164917 := proc(n) local p, a, j, q, itr ; p := ithprime(n) ; a := 1000000000000000 ; for j from 1 do q := A164368(j) ; if q > p then break; fi; itr := 0 ; while q < p do q := A060308(q) ; itr := itr+1 ; od; if q = p then if itr < a then a := itr; fi; fi; od: a ; end:
seq(A164917(n), n=1..120) ; # R. J. Mathar, Sep 24 2009
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CROSSREFS
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Cf. A006992 A104272 A164368 A164288
Sequence in context: A137663 A161628 A122059 this_sequence A166238 A014197 A021438
Adjacent sequences: A164914 A164915 A164916 this_sequence A164918 A164919 A164920
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KEYWORD
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nonn
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AUTHOR
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Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 31 2009
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EXTENSIONS
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Edited, examples added and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 24 2009
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