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A248601
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Square array T(n, k) read by antidiagonals, n > 0 and k > 0: for any number n > 0, let f(n) be the function that associates k to the prime(k)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the arithmetic functions with nonnegative integer values and a finite number of nonzero values; let g be the inverse of f; T(n, k) = g(f(n) * f(k)) (where i * j denotes the Dirichlet convolution of i and j).
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4
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 13, 16, 13, 6, 1, 1, 7, 21, 25, 25, 21, 7, 1, 1, 8, 19, 36, 23, 36, 19, 8, 1, 1, 9, 27, 49, 65, 65, 49, 27, 9, 1, 1, 10, 49, 64, 37, 126, 37, 64, 49, 10, 1, 1, 11, 39, 81, 125, 133, 133, 125
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OFFSET
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1,5
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COMMENTS
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For any n > 0, f(n) corresponds to the function k -> A249344(k, n).
For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
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); as such, f is an homomorphism from the multiplicative group of positive rational numbers to the additive group of arithmetic functions with integer values and a finite number of nonzero values.
For any arithmetic function with integer values and a finite number of nonzero values j, g(j) = Product_{k > 0} A000040(k)^j(k).
See A296857 for the main diagonal of T.
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LINKS
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FORMULA
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T is completely multiplicative in both parameters:
- for any n > 0
- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n * i)^e_i.
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(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(n, 3^i) = A297002(n)^i for any i >= 0,
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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
---+-------------------------------------------------
3| 1 3 7 9 13 21 19 27 49 39 -> A297002
4| 1 4 9 16 25 36 49 64 81 100 -> A000290
5| 1 5 13 25 23 65 37 125 169 115
6| 1 6 21 36 65 126 133 216 441 390
7| 1 7 19 49 37 133 53 343 361 259
8| 1 8 27 64 125 216 343 512 729 1000 -> A000578
9| 1 9 49 81 169 441 361 729 2401 1521
10| 1 10 39 100 115 390 259 1000 1521 1150
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PROG
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(PARI) T(n, k) = my(fn=factor(n), pn=apply(primepi, fn[, 1]~), fk=factor(k), pk=apply(primepi, fk[, 1]~)); prod(i=1, #pn, prod(j=1, #pk, prime(pn[i]*pk[j])^(fn[i, 2]*fk[j, 2])))
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CROSSREFS
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Cf. A000004, A000007, A000012 (first row/column), A000027 (second row/column), A000040, A000290 (fourth row/column), A000578 (eighth row/column), A001221, A001222, A055396, A061395, A249344, A289508, A296857 (main diagonal), A297002 (third row/column).
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KEYWORD
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AUTHOR
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STATUS
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approved
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