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A265673
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Area A under the trajectory of each odd number in the "3x+1" problem.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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:
The primes of the sequence are 47, 137, 509, 2161, 3967, 4007, ...
The squares of the sequence are 0, 4900, 8281, 61009, 176400, 181476,…
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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