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A177789 Irregular triangle in which row n gives the congruences (mod 2^A020914(n)) satisfied by the numbers having dropping time A122437(n+1) in the Collatz (3x+1) iteration. 1
0, 1, 3, 11, 23, 7, 15, 59, 39, 79, 95, 123, 175, 199, 219, 287, 347, 367, 423, 507, 575, 583, 735, 815, 923, 975, 999, 231, 383, 463, 615, 879, 935, 1019, 1087, 1231, 1435, 1647, 1703, 1787, 1823, 1855, 2031, 2203, 2239, 2351, 2587, 2591, 2907, 2975, 3119 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The dropping time is the number of Collatz iterations required to reach a lower number than starting value. Garner mentions these congruences. The first term in row n is A122442(n+1) for n>1. The length of row n is A100982(n). The triangle means:

numbers 0 (mod 2) and >0 have dropping time 1

numbers 1 (mod 4) and >1 have dropping time 3

numbers 3 (mod 16) have dropping time 6

numbers 11, 23 (mod 32) have dropping time 8

numbers 7, 15, 59 (mod 128) have dropping time 11

numbers 39, 79, 95, 123, 175, 199, 219 (mod 256) have dropping time 13

REFERENCES

Lynn E. Garner, On the Collatz 3n + 1 Algorithm, Proc. Amer. Math. Soc., Vol. 82(1981), 19-22.

Mike Winkler, On the structure and the behaviour of Collatz 3n+ 1 sequences, 2014; http://mikewinkler.co.nf/collatz_structure_2014.pdf

LINKS

Table of n, a(n) for n=0..50.

M. Winkler, On a stopping time algorithm of the 3n+ 1 function

M. Winkler, New results on the stopping time behaviour of the Collatz 3x + 1 function, arXiv:1504.00212 [math.GM], 2015.

MATHEMATICA

DroppingTime[n_] := Module[{m=n, k=0}, If[n>1, While[m>=n, k++; If[EvenQ[m], m=m/2, m=3*m+1]]]; k]; dt=Floor[1+Range[0, 20]*Log[2, 6]]; e=Floor[1+Range[0, 20]*Log[2, 3]]; Join[{0, 1}, Flatten[Table[Select[Range[3, 2^e[[n]], 2], DroppingTime[ # ]==dt[[n]] &], {n, 2, 8}]]]

CROSSREFS

Cf. A060445 (dropping time of odd numbers)

Sequence in context: A121471 A178946 A087078 * A289526 A289765 A141226

Adjacent sequences:  A177786 A177787 A177788 * A177790 A177791 A177792

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, May 13 2010

STATUS

approved

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.