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A300119
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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.
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1
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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
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OFFSET
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1,5
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COMMENTS
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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).
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LINKS
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Table of n, a(n) for n=1..84.
Rémy Sigrist, Scatterplot of (n, k) such that T(n, k) = 40 and 1 <= n <= 1000 and 1 <= k <= 1000
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
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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.
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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Cf. A006370, A006577.
Sequence in context: A003983 A087062 A204026 * A323211 A110537 A144434
Adjacent sequences: A300116 A300117 A300118 * A300120 A300121 A300122
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KEYWORD
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nonn,tabl
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AUTHOR
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Rémy Sigrist, Feb 25 2018
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STATUS
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approved
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