A334055 Iteration count (or -1) corresponding to A334054(n).

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 4, 10, 3, 1, 1, 1, 1, 3, 1, 3, 7, 2, 9, 3, 1, 7, 3, 3, 4, -1, 3, 3, -1, 4, 1, -1, 5, 9, -1, -1, 7, 3, 1, 2, 3, 1, 5, 9, 5, -1, 3, 1, 4, 3, 3, 4, -1, 3, 4, -1, 3, 1, -1, 5, 7, -1, 3, 1, 1, -1, 2, 9, 11, 3, 8, 4, 10, 3, 4, -1, -1, 3, -1, -1, 3, 3, -1, 8, 10

Offset: 1

Author

Scott R. Shannon, Sep 07 2020

Keywords

• sign

Comments

This is the number of iterations for the starting number, containing only digits 1,2 and 3 (see A007932) to reappear in the iterative cycle of its own 'Look and Say' description.

See A334054 for further details, a list of the number that do reappear in their iterative cycle, and a proof that the number 233 can never reappear in its cycle.

For all 1092 numbers up to six digits long containing digits 1,2,3 there are 397 numbers which reappear in their own iterative cycle, while 695 do not and can be proven never will.

In the same range, numbers 121322 and 213223 take 33 cycles before reappearing. The final string in the later case has 49470 digits. See the link file for details of the other numbers.

Examples

			The numbers containing only digits 1,2 and 3 are given in A007932.
a(1) = 1 as the number 1 take one iteration to reappear: 1->11 which contains '1' as a substring.
a(4) = 2 as the number 11 takes two iterations to reappear: 11->21->1211 which contains '11' as a substring.
a(9) = 3 as the number 23 takes three iterations to reappear: 23->1213->11121113->31123113 which contains '23' as a substring.
a(11) = 4 as the number 32 takes four iterations to reappear: 32->1312->11131112->31133112->1321232112 which contains '32' as a substring.
a(12) = 10 as the number 33 takes ten iterations to reappear: 33->23->1213->11121113->31123113->132112132113->11131221121113122113->311311222112311311222113->1321132132211213211321322113->11131221131211132221121113122113121113222113->3113112221131112311332211231131122211311123113322113 which contains '33' as a substring.
		

Links

• Scott R. Shannon, Details of all 1092 numbers up to six digits long.

Crossrefs

Cf. A334054, A005150, A007932, A119566, A007932, A045918.