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A324592
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Square array T(n, k) read by diagonals, n > 0, k > 0; for any number m > 0 with prime factorization Product_{i > 0} prime(i)^e(i), let f(m) = Sum_{i > 0} e(i) * sqrt(A005117(i)); f establishes a bijection between the positive numbers and the finite sums of square roots of squarefree numbers; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
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1
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 11, 16, 11, 6, 1, 1, 7, 12, 25, 25, 12, 7, 1, 1, 8, 17, 36, 8, 36, 17, 8, 1, 1, 9, 27, 49, 55, 55, 49, 27, 9, 1, 1, 10, 16, 64, 31, 72, 31, 64, 16, 10, 1, 1, 11, 33, 81, 125, 119, 119, 125
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OFFSET
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1,5
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COMMENTS
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The set of square roots of squarefree numbers, { sqrt(A005117(i)), i > 0 }, is Q-linearly independent. The set of finite sums of square roots of squarefree numbers is closed under multiplication, hence the sequence is well defined.
The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v).
This sequence has similarities with A297845.
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LINKS
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FORMULA
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For any m > 0, n > 0 and k > 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(m, n*k) = T(m, n) * T(m, k) and T(n*k, m) = T(n, m) * T(k, m) (T is completely multiplicative in both parameters),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 2^i) = n^i for any i >= 0,
- T(prime(n), prime(n)) = 2^A005117(n) (where prime(n) denotes the n-th prime number).
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EXAMPLE
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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 3 4 5 6 7 8 9 10
3| 1 3 4 9 11 12 17 27 16 33
4| 1 4 9 16 25 36 49 64 81 100
5| 1 5 11 25 8 55 31 125 121 40
6| 1 6 12 36 55 72 119 216 144 330
7| 1 7 17 49 31 119 32 343 289 217
8| 1 8 27 64 125 216 343 512 729 1000
9| 1 9 16 81 121 144 289 729 256 1089
10| 1 10 33 100 40 330 217 1000 1089 400
For n = 3 and k = 5:
- f(3) = f(prime(2)) = sqrt(A005117(2)) = sqrt(2),
- f(5) = f(prime(3)) = sqrt(A005117(3)) = sqrt(3),
- f(3) * f(5) = sqrt(6) = sqrt(A005117(5)),
- hence T(3, 5) = prime(5) = 11.
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PROG
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(PARI) See Links section.
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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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