A350805 - OEIS

%I #17 Jan 18 2022 05:24:23

%S 0,1,35,2,4,11,31,36,9,34,10,1,19,32,5,3,33,7,17,8,6,30,31,20,6,5,2,

%T 21,9,2,30,5,30,9,30,30,7,30,30,18,5,31,30,30,31,38,38,33,31,30,33,32,

%U 8,30,31,7,30,16,30,22,36,32,32,12,8,32,34,30,10,30,37,32,34,30,32,23,8,34,34

%N Number of iterations to reach 1 or a repeated value when starting at n for the map: x -> Sum_{prime factors of x} A006370(x), where A006370 is the Collatz 3x+1 map.

%C The map sums the values of the Collatz 3x+1 map for each of the prime factors of the input value. Therefore any prime factor 2 adds one to the sum while any other prime factor p adds 3*p+1. See the Examples below.

%C An examination of the first 10^9 terms shows about 22.8% of all starting values go to 1. The majority, about 68.2%, go into a thirty-member loop with a lowest value 22 and a highest value 1385. About 7.5% go into a five-member loop with lowest value 15 and highest value 124, while about 1.4% go into a two-member loop with values 27 and 30. The remaining terms, about 0.009%, go to 12 which maps to itself. It is possible these four loops are the only ones for all starting values, although this is unknown. See A350807 for the loop values.

%C In the same range the starting value with the most iterations before reaching 1 or repeating is 622074454, which goes into the thirty-member loop and repeats 22 after sixty-three iterations. See A350806 for the largest value reached starting from n. The starting value with the largest ratio of maximum value reached to starting value is 19762559 which reaches 24311654278 after nine iterations, a ratio of about 1230. Note that after three more iterations its value has reduced to 95.

%e a(2) = 1 as 2 -> 1 in the 3x+1 map, which then has no further prime factors.

%e a(3) = 35 as 3 -> 10 -> 17 -> 52 -> 42 -> 33 -> 44 -> 36 -> 22 -> 35 -> 38 -> 59 -> 178 -> 269 -> 808 -> 307 -> 922 -> 1385 -> 848 -> 164 -> 126 -> 43 -> 130 -> 57 -> 68 -> 54 -> 31 -> 94 -> 143 -> 74 -> 113 -> 340 -> 70 -> 39 -> 50 -> 33, which repeats (33) after thirty-five iterations. This is the first number to enter the thirty-member loop.

%e a(5) = 4 as 5 -> 16 -> 4 -> 2 -> 1.

%e a(12) = 1 as 12 -> 12. The prime factorization of 12 is 2*2*3, which maps to 1,1,10, and 1+1+10 = 12. This is the only known fixed point/one-term loop.

%e a(15) = 5 as 15 -> 26 -> 41 -> 124 -> 96 -> 15, which repeats (15) after five iterations. This is the first number to enter the five-member loop.

%e a(27) = 2 as 27 -> 30 -> 27, which repeats (27) after two iterations. This is the first number to enter the two-member loop.

%e a(4096) = 2 as 4096 -> 12 -> 12, which repeats (12) after two iterations. This is the first value, other than 12, that ends at 12.

%Y Cf. A006370, A027746, A350806, A350807, A006577, A000040.

%K nonn

%O 1,3

%A _Scott R. Shannon_, Jan 17 2022