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A138754
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PrimePi(A138751(n)) - a variation of the Collatz (3n+1) map.
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5
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1, 4, 2, 7, 4, 10, 5, 13, 6, 7, 19, 22, 9, 24, 10, 10, 11, 31, 33, 12, 35, 38, 14, 15, 45, 16, 47, 17, 48, 17, 55, 19, 20, 60, 22, 63, 66, 67, 24, 24, 25, 73, 25, 77, 26, 79, 83, 87, 31, 89, 31, 31, 93, 31, 32, 33, 33, 101, 102, 35, 104, 35, 113, 37, 115, 38, 122, 123, 41, 126
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| This map is a variation of the Collatz (or 3n+1) map:
Instead of considering the parity of the number, we look at
prime(n) (mod 3) to decide if this prime should be halved or doubled,
before going to the next prime (A007918) and finally back to the
positive integers via PrimePi (A000720).
Exactly like for the Collatz (3n+1) map (defined on nonnegative integers),
the first element for which it is defined is its only fixed point,
and all other starting values seem to end up in a cycle of length 3,
here: 4 -> 7 -> 5 -> 4.
Except for p=3, no prime yields a prime result under the
map A138750 (as can be seen using p=6k+1 or p=6k-1). Therefore
instead of applying primepi() after nextprime(), one could also simply use 1+primepi().
The prime p=3 is also the only case where n=2(mod 3) is not equivalent
to n != 1 (mod 3). It might have been a better choice to define
A138750(x)=2x if x=1 mod 3, =ceil(x/2) else. But since here it makes
only a difference for p=3, we use the original definition (cf
A124123).
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LINKS
| Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
| A138754(n) = A000720(A138751(n)) = A000720(A007918(A138750(A000040(n))))
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EXAMPLE
| a(4) = 7 since prime(4) = 7 = 1 (mod 3), thus A138750(7) =
2*7 = 14, nextprime(14) = 17, PrimePi(17) = 7 (i.e. 17 is the 7-th
prime).
a(5) = 4 since prime(5) = 11 = 2 (mod 3), thus A138750(11)
= ceil(11/2) = 6, nextprime(6) = 7, PrimePi(7) = 4 (i.e. 7 is the 4-th
prime).
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PROG
| (PARI) A138754(n)=primepi(A138751(n)) /* see there */
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CROSSREFS
| Cf. A124123, A138750-A138753, A000040, A000720, A007918.
Sequence in context: A197026 A195779 A130849 * A021963 A131914 A115302
Adjacent sequences: A138751 A138752 A138753 * A138755 A138756 A138757
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KEYWORD
| easy,nonn
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AUTHOR
| M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 01 2008
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