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A078350
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Number of primes in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function defined by f(x) = x/2 if x is even; f(x) = 3x + 1 if x is odd.
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7
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0, 1, 3, 1, 2, 3, 6, 1, 6, 2, 5, 3, 3, 6, 4, 1, 4, 6, 7, 2, 1, 5, 4, 3, 7, 3, 25, 6, 6, 4, 24, 1, 7, 4, 3, 6, 7, 7, 11, 2, 25, 1, 8, 5, 4, 4, 23, 3, 7, 7, 6, 3, 3, 25, 24, 6, 8, 6, 11, 4, 5, 24, 20, 1, 7, 7, 9, 4, 3, 3, 22, 6, 25, 7, 2, 7, 6, 11, 11, 2, 5, 25, 24, 1, 1, 8, 9, 5, 10, 4, 20, 4, 3, 23, 20
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Number of primes in the trajectory of n under the 3x+1 map (i.e. the number of primes until the trajectory reaches 1, including 2 once). - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 23 2002
a(A196871(n)) = 0. [Reinhard Zumkeller, Oct 08 2011]
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Eric Weisstein's World of Mathematics,
CollatzProblem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
| Conjecture : for n>A a(n)>Log(n)/Log(Log(n)) (A>10^6) - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 23 2002
a(n) = A055509(n) + 1 for n > 1.
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EXAMPLE
| 3 ->10 ->5 ->16 ->8 ->4 ->2 ->1 so in this trajectory 3,5,2 are primes hence a(3)=3. - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 23 2002
The finite sequence n, f(n), f(f(n)), ...., 1 for n = 12 is: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, which has three prime terms. Hence a(12) = 3.
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MATHEMATICA
| f[n_] := n/2 /; Mod[n, 2] == 0 f[n_] := 3 n + 1 /; Mod[n, 2] == 1 g[n_] := Module[{i, p}, i = n; p = 0; While[i > 1, If[PrimeQ[i], p = p + 1]; i = f[i]]; p]; Table[g[n], {n, 1, 100}]
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PROG
| (PARI) for(n=2, 500, s=n; t=0; while(s!=1, if(isprime(s)==1, t=t+1, t=t); if(s%2==0, s=s/2, s=(3*s+1)); if(s==1, print1(t, ", "); ); )) - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 23 2002
(Haskell)
a078350 n = sum $ map a010051 $ takeWhile (> 1) $ iterate a006370 n
-- Reinhard Zumkeller, Oct 08 2011
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CROSSREFS
| Cf. A064684.
Cf. A006370, A010051.
Sequence in context: A107341 A138881 A070983 * A078719 A087227 A060477
Adjacent sequences: A078347 A078348 A078349 * A078351 A078352 A078353
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KEYWORD
| nice,nonn
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AUTHOR
| Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 23 2002
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 17 2009 at the suggestion of R. J. Mathar
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