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A078440 Numbers n with property that n is not a power of 2 and the finite sequence n, f(n), f(f(n)), ...., 1 in the Collatz (or 3x + 1) problem contains exactly one prime. (The earliest "1" is meant.) 2
21, 42, 84, 85, 168, 170, 336, 340, 341, 453, 672, 680, 682, 906, 909, 1344, 1360, 1364, 1365, 1812, 1813, 1818, 2688, 2720, 2728, 2730, 3624, 3626, 3636, 5376, 5440, 5456, 5460, 5461, 7248, 7252, 7272, 7281, 9669 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

f(n) = n/2 if n is even, = 3n + 1 if n is odd. Powers 2^n trivially have exactly one prime in n, f(n), f(f(n)), ..., 2, 1, namely 2 and so are excluded from the sequence.

A055509(a(n)) = 0; A078350(a(n)) <= 1.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

Index entries for sequences related to 3x+1 (or Collatz) problem

EXAMPLE

n, f(n), f(f(n)), .... for n = 21 is: 21, 64, 32, 16, 8, 4, 2, 1, which has exactly one prime, that is, 2. Hence 21 belongs to the sequence.

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]; Select[Range[10^4], g[ # ] == 1 && ! IntegerQ[Log[2, # ]] &]

pQ[n_]:=Count[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&], _?PrimeQ] == 1; With[ {nn=10000}, Complement[Select[Range[nn], pQ], 2^Range[Floor[ Log[ 2, nn]]]]] (* Harvey P. Dale, Oct 19 2011 *)

PROG

(Haskell)

a078440 n = a078440_list !! (n-1)

a078440_list = filter notbp a196871_list where

   notbp x = m > 0 && x > 1 || m == 0 && notbp x' where

      (x', m) = divMod x 2

-- Reinhard Zumkeller, Oct 08 2011

CROSSREFS

A006370; subsequence of A196871 (with binary powers).

Sequence in context: A086794 A001682 A180963 * A175805 A039344 A043167

Adjacent sequences:  A078437 A078438 A078439 * A078441 A078442 A078443

KEYWORD

nonn

AUTHOR

Joseph L. Pe, Dec 31 2002

STATUS

approved

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Last modified November 26 03:20 EST 2020. Contains 338632 sequences. (Running on oeis4.)