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 A336913 Image of n under the 3^x+1 map, which is a variation of the 3x+1 (Collatz) map. 1
 4, 1, 28, 2, 244, 2, 2188, 3, 19684, 3, 177148, 3, 1594324, 3, 14348908, 4, 129140164, 4, 1162261468, 4, 10460353204, 4, 94143178828, 4, 847288609444, 4, 7625597484988, 4, 68630377364884, 4, 617673396283948, 5, 5559060566555524, 5, 50031545098999708, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It seems that all 3^x+1 trajectories reach 1; this has been verified up to 10^9. Once a 3^x+1 trajectory reaches 1, it repeats the following cycle: 1, 4, 2, 1, 4, 2, 1, ... LINKS Wikipedia, Collatz conjecture FORMULA a(n) = floor(log_2(n)) if n is even, 3^n+1 if n is odd. EXAMPLE For n = 5, a(5) = 3^5+1 = 244, because 5 is odd. For n = 6, a(6) = floor(log_2(6)) = 2, because 6 is even. PROG (Python) from math import floor, log def a(n): return 3**n + 1 if n % 2 else int(floor(log(n, 2))) print([a(n) for n in range(1, 51)]) (Python) ''' Program that confirms that 3^x+1 trajectories end with 1. We avoid the expensive 3^n+1 calculation based on the following: - 3^n is not a power of two (for n >= 1). - 3^n+1 is not a power of two (for n > 1) because of the Catalan Conjecture, which was proven in 2002. - Thus, floor(log2(3^n+1)) == floor(log2(3^n)) == floor(n*log2(3)) for n > 1. Thanks to Clark R. Lyons for this optimization. ''' from math import floor, log log2_of_3 = log(3, 2) # 16 digits after the decimal point. max_n = 10**15 / 2    # Larger values multiplied by log2_of_3 may have rounding errors. def check_trajectory(n):     while n > 1:         if n % 2 == 0:             n = int(floor(log(n, 2)))         else:             if n > max_n:                 raise ValueError(str(n) + " is too large to be multiplied by log2_of_3")             n = int(floor(n * log2_of_3)) n = 1 while n <= 1000000000:     check_trajectory(n)     n += 1 CROSSREFS Cf. A006370 (image of n under the 3x+1 map). Cf. A336914 (gives number of steps to reach 1). See also A199561. Sequence in context: A139051 A061692 A096206 * A134150 A134151 A264773 Adjacent sequences:  A336910 A336911 A336912 * A336914 A336915 A336916 KEYWORD nonn AUTHOR Robert C. Lyons, Aug 08 2020 STATUS approved

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Last modified July 31 02:33 EDT 2021. Contains 346367 sequences. (Running on oeis4.)