A248601 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).

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

Offset: 1

Author

Rémy Sigrist, Dec 21 2017

Keywords

• nonn
• tabl
• mult

Comments

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).

Also, f(1) = A000004 and f(2) corresponds to k -> A000007(k-1).

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.

Examples

			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  -> A000012
    2|  1   2   3    4    5    6    7     8     9    10  -> A000027
    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
		

Links

• Wikipedia, Dirichlet convolution.

Crossrefs

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).

Programs

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])))

Formula

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, 4) = n^2 (A000290),

- T(n, 8) = n^3 (A000578),

- T(n, 3) = A297002(n),

- T(n, 3^i) = A297002(n)^i for any i >= 0,

- A001221(T(n, k)) <= A001221(n) * A001221(k),

- A001222(T(n, k)) = A001222(n) * A001222(k),

- A055396(T(n, k)) = A055396(n) * A055396(k),

- A061395(T(n, k)) = A061395(n) * A061395(k),

- A289508(T(n, k)) = A289508(n) * A289508(k),

- T(A000040(n), A000040(k)) = A000040(n * k),

- T(A000040(n)^i, A000040(k)^j) = A000040(n * k)^(i * j) for any i >= 0 and j >= 0.