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A265673 Area A under the trajectory of each odd number in the "3x+1" problem. 1
0, 47, 33, 284, 334, 253, 112, 686, 205, 509, 137, 621, 645, 101426, 427, 101088, 824, 522, 516, 2324, 101310, 1316, 474, 100955, 781, 1104, 372, 102288, 1624, 2157, 985, 105066, 1132, 1425, 543, 100754, 102682, 1482, 886, 4344, 1422, 102053, 553, 2672, 1905 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For each odd number, we consider the 3x+1 function iterates (x, T(x), T(T(x)),...,4,2,1) plotted on standard vertical and horizontal scale. Each point (0,x), (1,T(x)), (2, T(T(x)),...,(r,1), where r is the number of iterations needed to reach 1, is connected by a straight line. The area A is the sum of trapezes of base b = 1, such that A =(x + T(x))/2 + (T(x) + T(T(x)))/2 + ... + 3/2.

The area A is an integer number if x is odd because A = ((x+1) + 2s)/2 = (x+1)/2 + s with s = T(x) + T(T(x) + T(T(T(x))) + ... + 4 + 2.

The area is not unique for different values of n. Examples:

a(63) = a(2342) = 101699;

a(174) = a(450) = 118514;

a(241) = a(332) = 17225;

a(563) = a(867) = 13787;

a(697) = a(1131) = 14991;

a(1178) = a(1426) = 44427.

There exists index i, j such that a(i) = a(j) with trajectories having the same length, this implies that A006577(2i-1) = A006577(2j-1) where A006577 is the number of halving and tripling steps to reach 1 in "3x+1" problem. Examples:

a(2696) = a(17195) = 231417 and A006577(5391) = A006577(34389) = 28;

a(5485) = a(6025) = 183649 and A006577(10969) = A006577(12049) = 42;

a(9137) = a(9297) = 950016  and A006577(18273) = A006577(18593) = 154.

The primes of the sequence are  47, 137, 509, 2161, 3967, 4007, ...

The squares of the sequence are  0, 4900, 8281, 61009, 176400, 181476,…

LINKS

Michel Lagneau, Table of n, a(n) for n = 1..10000

EXAMPLE

a(2) = 47 because the trajectory of the second odd number 3 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1, and the area A = (3+10)/2 + (10+5)/2 + (5+16)/2 +(16+8)/2 + (8+4)/2 +(4+2)/2 +(2+1)/2 = 94/2 = 47.

MAPLE

nn:=2000:for k from 1 to 100 do:n:=2*k-1:s:=0:m:=n:for i from 1 to nn while(m<>1) do:if irem(m, 2)=0 then s:=s+(m+m/2)/2:m:=m/2:else s:=s+(m+3*m+1)/2:m:=3*m+1:fi:od:printf(`%d, `, s):od:

MATHEMATICA

lst={}; f[n_]:=Module[{s=0, a=2*n-1, k=0}, While[a!=1, k++; If[EvenQ[a], s=s+(a+a/2)/2; a=a/2, s=s+(a+a*3+1)/2; a=a*3+1]]; k; AppendTo[lst, s]]; Table[f[n], {n, 100}]; lst

CROSSREFS

Cf. A006577.

Sequence in context: A217423 A033367 A052352 * A089553 A165868 A291513

Adjacent sequences:  A265670 A265671 A265672 * A265674 A265675 A265676

KEYWORD

nonn

AUTHOR

Michel Lagneau, Dec 13 2015

STATUS

approved

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Last modified February 22 12:00 EST 2018. Contains 299452 sequences. (Running on oeis4.)