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A349954 a(n) is the number of extrema that result from iterating the reduced Collatz function R(k) = A139391(k) on 2n-1 to yield 1. 1
0, 2, 1, 2, 3, 2, 1, 2, 1, 4, 1, 2, 5, 20, 3, 18, 5, 2, 3, 8, 19, 4, 1, 18, 3, 4, 1, 20, 5, 8, 3, 18, 3, 6, 1, 18, 21, 2, 3, 6, 3, 20, 1, 4, 7, 16, 3, 18, 21, 4, 5, 14, 7, 18, 19, 10, 1, 4, 3, 6, 17, 12, 19, 4, 21, 4, 5, 6, 15, 10, 1, 18, 19, 22, 3, 2, 5, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The trajectory starts with a minimum for odd n and with a maximum (see A351974) for even n (>=2). Since the trajectory always stops at 1 (a minimum) assuming the Collatz conjecture holds, a(n) is odd if n is odd and vice versa.

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

a(10) = 4 because 2n+1 = 19 and iterating R on 19 gives 4 extrema:

19 -> 29 -> 11 -> 17 -> 1

max min max min.

The corresponding path of n, 10 -> 15 -> 6 -> 9 -> 1, is shown in the tree below, where the paths for n up to 100 are given and a(n) is the depth from n to 1.

n a(n)

----------------------------------------------------------------------------- ----

98 74 22

37 49 147 65 111 21

14 86 \__\__28_/ 42 100 20

95 21 55 73 83 97 129 63_____/ 225 19

54 36 \___\__\__\___\__16 24 48 32 72 18

\__\____________________\________81 61 243__/__/ 17

\______\___46 92 16

69 207 15

52 78 14

117__/ 13

62 88 12

93 297 11

70 94 84 56 10

105 79 141 189__/ 9

20 30__/ 106 142 8

\__45 159 53 213 7

68 34 60 40 90 160 80 6

29 153 77 85 13 51 17 67 89 135_/___/ 1215 405 5

\__22 50 58 44 66 26 64 96 \__10__/__/__/__/ 82 456 304 4

5 19 25 33 75 87 99_/ 39 729_/ 59 15 47 123 1539__/ 31 41 3

\__\__\___\__\__\__4 \___6____/___/ 76 38 2 8 18 \___12_____/__/ 2

\_________9 11 43 71 171 57 3 \__\_______27 91 35 23 7 1

\__\__\___\___\__\__\_______________1__/__/__/__/ 0

PROG

(Python)

def R(k): c = 3*k+1; return c//(c&-c)

def A349954(n):

if n == 1: return 0

ct = 1; m = R(2*n-1); d = m - 2*n + 1

while m > 1:

if (R(m) - m)*d < 0: ct += 1; d = -d

m = R(m)

return ct

CROSSREFS

Cf. A075677, A075680, A122458, A139391, A256598, A351974.

Sequence in context: A317952 A059131 A059129 * A081771 A338170 A066856

Adjacent sequences: A349951 A349952 A349953 * A349955 A349956 A349957

KEYWORD

nonn

AUTHOR

Ya-Ping Lu, Mar 11 2022

STATUS

approved

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Last modified February 7 01:07 EST 2023. Contains 360111 sequences. (Running on oeis4.)