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A287957
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Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.
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3
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1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1
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OFFSET
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1,5
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COMMENTS
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We use the definition of recursive divisor given in A282446.
More informally, the prime tower factorization of T(n, k) is the intersection of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
This sequence has connections with the classical GCD (A003989).
For any i > 0, j > 0 and k > 0:
- T(i, j) = 1 iff gcd(i, j) = 1,
- T(i, j) >= 1,
- T(i, j) <= min(i, j),
- T(i, j) <= gcd(i, j),
- T(i, 1) = 1,
- T(i, i) = i,
- T(i, j) = T(j, i) (the sequence is commutative),
- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
- T(i, i*j) <= i,
- if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),
See also A287958 for the LCM equivalent.
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LINKS
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EXAMPLE
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Table starts:
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 1 2 1 2 1 2 1 2 ...
3 | 1 1 3 1 1 3 1 1 3 1 ...
4 | 1 2 1 4 1 2 1 2 1 2 ...
5 | 1 1 1 1 5 1 1 1 1 5 ...
6 | 1 2 3 2 1 6 1 2 3 2 ...
7 | 1 1 1 1 1 1 7 1 1 1 ...
8 | 1 2 1 2 1 2 1 8 1 2 ...
9 | 1 1 3 1 1 3 1 1 9 1 ...
10 | 1 2 1 2 5 2 1 2 1 10 ...
...
T(4, 8) = T(2^2, 2^3) = 2.
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PROG
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(PARI) T(n, k) = my (g=factor(gcd(n, k))); return (prod(i=1, #g~, g[i, 1]^T(valuation(n, g[i, 1]), valuation(k, g[i, 1]))))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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