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%I #9 Feb 26 2018 19:22:22
%S 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,4,4,2,1,1,2,5,4,5,2,1,1,2,3,4,4,3,
%T 2,1,1,2,10,4,5,4,10,2,1,1,2,8,4,5,5,4,8,2,1,1,2,10,4,5,6,5,4,10,2,1,
%U 1,2,10,4,8,10,10,8,4,10,2,1,1,2,10,4,5,8
%N Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is the value where Collatz sequences starting at n and at k meet.
%C For any n > 0 and k > 0, if the Collatz conjecture is true, then:
%C - A006370^i(n) = 1 and A006370^j(k) = 1 for some i >= 0 and j >= 0 (where A006370^i denotes the i-th iterate of A006370; actually i = A006577(n) and j = A006577(k)),
%C - hence the Collatz sequences starting at n and k meet,
%C - let c be the greatest number between 0 and min(i, j) inclusive such that A006370^(i-c)(n) = A006370^(j-c)(k),
%C - then T(n, k) = A006370^(i-c)(n).
%H Rémy Sigrist, <a href="/A300119/a300119.png">Scatterplot of (n, k) such that T(n, k) = 40 and 1 <= n <= 1000 and 1 <= k <= 1000</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F For any m > 0, n > 0 and k > 0, and provided that the Collatz conjecture is true:
%F - T(n, n) = n,
%F - T(n, k) = T(k, n) (T is commutative),
%F - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F - T(n, 1) = 1 (1 is an absorbing element for T),
%F - T(n * 2^k, n) = n,
%F - T(n, k) = 1 iff n = 1 or k = 1.
%e For T(12, 13):
%e - The Collatz sequence starting at 12 is: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1,
%e - The Collatz sequence starting at 13 is: 13, 40, 20, 10, 5, 16, 8, 4, 2, 1,
%e - They meet at the value 10, hence T(12, 13) = 10.
%e Array T(n, k) begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10
%e ---+--------------------------------------------------
%e 1| 1 1 1 1 1 1 1 1 1 1
%e 2| 1 2 2 2 2 2 2 2 2 2
%e 3| 1 2 3 4 5 3 10 8 10 10
%e 4| 1 2 4 4 4 4 4 4 4 4
%e 5| 1 2 5 4 5 5 5 8 5 5
%e 6| 1 2 3 4 5 6 10 8 10 10
%e 7| 1 2 10 4 5 10 7 8 7 10
%e 8| 1 2 8 4 8 8 8 8 8 8
%e 9| 1 2 10 4 5 10 7 8 9 10
%e 10| 1 2 10 4 5 10 10 8 10 10
%o (PARI) T(n,k) = my (nn=[]); while (1, nn = concat(nn,n); if (n==1, break); n=if (n%2, 3*n+1, n/2)); nn=Set(nn); while (!setsearch(nn,k), k=if (k%2, 3*k+1, k/2)); k
%Y Cf. A006370, A006577.
%K nonn,tabl
%O 1,5
%A _Rémy Sigrist_, Feb 25 2018
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