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

1,2

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.

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.

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).

Example

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.

Prog

(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])))

Crossrefs

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

Sequence in context: A041573 A211053 A041341 * A229595 A083508 A048231

Adjacent sequences: A296854 A296855 A296856 * A296858 A296859 A296860

Keyword

nonn

Author

Rémy Sigrist, Dec 21 2017

Status

approved