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 A300119 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. 1
 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, 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, 1, 2, 10, 4, 8, 10, 10, 8, 4, 10, 2, 1, 1, 2, 10, 4, 5, 8 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS For any n > 0 and k > 0, if the Collatz conjecture is true, then: - 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)), - hence the Collatz sequences starting at n and k meet, - let c be the greatest number between 0 and min(i, j) inclusive such that A006370^(i-c)(n) = A006370^(j-c)(k), - then T(n, k) = A006370^(i-c)(n). LINKS FORMULA For any m > 0, n > 0 and k > 0, and provided that the Collatz conjecture is true: - T(n, n) = n, - T(n, k) = T(k, n) (T is commutative), - T(m, T(n, k)) = T(T(m, n), k) (T is associative), - T(n, 1) = 1 (1 is an absorbing element for T), - T(n * 2^k, n) = n, - T(n, k) = 1 iff n = 1 or k = 1. EXAMPLE For T(12, 13): - The Collatz sequence starting at 12 is: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, - The Collatz sequence starting at 13 is: 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, - They meet at the value 10, hence T(12, 13) = 10. Array T(n, k) begins: n\k| 1 2 3 4 5 6 7 8 9 10 ---+-------------------------------------------------- 1| 1 1 1 1 1 1 1 1 1 1 2| 1 2 2 2 2 2 2 2 2 2 3| 1 2 3 4 5 3 10 8 10 10 4| 1 2 4 4 4 4 4 4 4 4 5| 1 2 5 4 5 5 5 8 5 5 6| 1 2 3 4 5 6 10 8 10 10 7| 1 2 10 4 5 10 7 8 7 10 8| 1 2 8 4 8 8 8 8 8 8 9| 1 2 10 4 5 10 7 8 9 10 10| 1 2 10 4 5 10 10 8 10 10 PROG (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 CROSSREFS Cf. A006370, A006577. Sequence in context: A003983 A087062 A204026 * A323211 A110537 A144434 Adjacent sequences: A300116 A300117 A300118 * A300120 A300121 A300122 KEYWORD nonn,tabl AUTHOR Rémy Sigrist, Feb 25 2018 STATUS approved

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Last modified February 7 22:58 EST 2023. Contains 360132 sequences. (Running on oeis4.)