

A260590


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


5



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
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OFFSET

1,1


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 nvalues are the indices of odd numbers (A005408).


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. 2021, 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".


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).


EXAMPLE

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.


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]


CROSSREFS

Cf. A005408, A006577, A020857, A075677, A075884, A076536, A144396, A166245.
Sequence in context: A286842 A087056 A076129 * A010648 A272335 A198994
Adjacent sequences: A260587 A260588 A260589 * A260591 A260592 A260593


KEYWORD

nonn


AUTHOR

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


STATUS

approved



