A296857 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; a(n) = g(f(n) * f(n)) (where i * j denotes the Dirichlet convolution of i and j).

1, 2, 7, 16, 23, 126, 53, 512, 2401, 1150, 97, 9072, 151, 5194, 27209, 65536, 227, 388962, 311, 230000, 133931, 23474, 419, 2612736, 279841, 51038, 40353607, 2036048, 541, 12244050, 661, 33554432, 571039, 131206, 1668811, 252047376, 827, 224542, 1447033

Offset: 1

Author

Rémy Sigrist, Dec 21 2017

Keywords

• nonn

Comments

This sequence is the main diagonal of A248601.

See A248601 for additional comments.

For any n > 0, gcd(2 * n, a(2 * n)) = 2 * n.

Examples

			For n = 12:
- f(12) = (2, 1, 0, 0, ...),
- f(12) * f(12) = (4, 4, 0, 1, 0, 0, ...),
- a(12) = prime(1)^4 * prime(2)^4 * prime(4) = 2^4 * 3^4 * 7 = 9072.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 50000 terms (where the color is function of A001222(n))
• Wikipedia, Dirichlet convolution.

Crossrefs

Cf. A000040, A001221, A001222, A055396, A061395, A248601.

Programs

PARI

a(n) = my (f=factor(n), p=apply(primepi, f[,1]~)); prod(i=1, #p, prod(j=1, #p, prime(p[i]*p[j])^(f[i,2]*f[j,2])))

Formula

For any n > 0 and k >= 0:

- a(n) = A248601(n, n),

- A001221(a(n)) <= A001221(n)^2,

- A001222(a(n)) = A001222(n)^2,

- A055396(a(n)) = A055396(n)^2,

- A061395(a(n)) = A061395(n)^2,

- a(A000040(n)) = A011757(n),

- a(A000040(n)^k) = A011757(n)^(k^2).