A260590 The modified Syracuse algorithm, msa, applied to 2n+1.

4, 2, 7, 2, 5, 2, 7, 2, 4, 2, 5, 2, 59, 2, 56, 2, 4, 2, 8, 2, 5, 2, 54, 2, 4, 2, 5, 2, 7, 2, 54, 2, 4, 2, 51, 2, 5, 2, 8, 2, 4, 2, 5, 2, 45, 2, 8, 2, 4, 2, 42, 2, 5, 2, 31, 2, 4, 2, 5, 2, 8, 2, 15, 2, 4, 2, 7, 2, 5, 2, 7, 2, 4, 2, 5, 2, 40, 2, 21, 2, 4, 2, 29, 2, 5, 2, 8, 2, 4, 2, 5, 2, 7, 2, 13

Offset: 1

Author

Joseph K. Horn and Robert G. Wilson v, Jul 29 2015

Keywords

• nonn

Comments

Normally the '3x+1 problem' or 'Collatz problem' asks for the number of steps to go from n to 1 (A006577). Here we ask for the number of iterations of the mapping, msa, to go from n to less than n; the mapping of x is either -> (3x+1)/2 if x is odd or -> x/2 if x is even.

Since the number of iterations of msa for an even number is always 1, we will only investigate the odd numbers greater than one.

a(n) = 1 for no values of n;

a(n) = 2 for n = 2 + 2k (k=0,1,2,3,...);

a(n) = 3 for no values of n;

a(n) = 4 for n = 1 + 8k (k=0,1,2,3,...);

a(n) = 5 for n = 5 + 16k and 11 + 16k (k=0,1,2,3,...);

a(n) = 6 for no values of n;

a(n) = 7 for n = 3 + 64k, 7 + 64k, 29 + 64k, etc. (k=0,1,2,3,...).

Possible values for a(n) are: 2, 4, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, ... (A260593, sorted). Density is ~ 5/8.

Record values: 4, 7, 59, 81, 105, 135, 164, 165, 173, 176, 183, 224, 246, 287, 292, 298, 308, 376, 395, 398, 433, 447, 547, ....

And the records occur for n: 1, 3, 13, 351, 5043, 17827, 135135, 181171, 190863, 313165, 513715, 563007, 4044031, 6710835, 10319167, 13358335, 28462477, 31864063, 108870007, 600495895, 913698783, 1394004493, ....

Remember these n-values are the indices of odd numbers (A005408).

Examples

			a(1) is 4 because 2n+1 is 3 and 3 -> 5 -> 8 -> 4 -> 2. The number of iterations of the msa is 4;
a(2) is 2 because 2n+1 is 5 and 5 -> 8 -> 4. The number of iterations of the msa is 2;
a(3) is 7 because 2n+2 is 7 and 7 -> 11 -> 17 -> 26 -> 13 -> 20 -> 10 -> 5. The number of iterations of the msa is 7; etc.
Also see The Modified Syracuse Algorithm link.
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 1..10000
• Encyclopedia of Mathematics, Syracuse problem.
• Joseph K. Horn, HHC 2014, HP Handheld Conference, Sept. 20-21, 2014, Reno, NV, Hailstone Numbers: A Pattern Has Been Found.
• Joseph K. Horn, The Modified Syracuse Algorithm.
• Eric Weisstein's World of Mathematics, The Syracuse Algorithm
• Wikipedia, Collatz conjecture. particularly Section "Cycles".

Crossrefs

Cf. A005408, A006577, A020857, A075677, A075884, A076536, A144396, A166245.

Programs

Mathematica

msa[n_] := If[ OddQ@ n, (3n + 1)/2, n/2]; f[n_] := Block[{k = 2n + 1}, Length@ NestWhileList[ msa@# &, k, # >= k &] - 1]; Array[f, 95]

Formula

a(n) = the number of iterations for the msa; i.e., the number of mappings of x -> (3x+1)/2 if x is odd or -> x/2 if x is even to arrive at a number less than n.

a(n) = the binary length of A260592(n).