A297473 For any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; a(n) = g(f(n)^2).

1, 2, 5, 16, 11, 90, 17, 512, 625, 550, 23, 6480, 31, 1666, 2695, 65536, 41, 101250, 47, 110000, 10285, 5566, 59, 1866240, 14641, 10478, 1953125, 653072, 67, 1212750, 73, 33554432, 19435, 23698, 31603, 65610000, 83, 33934, 44795, 88000000, 97, 9071370, 103

Offset: 1

Author

Rémy Sigrist, Dec 30 2017

Keywords

• nonn

Comments

This sequence is the main diagonal of A297845.

This sequence has similarities with A296857.

Examples

			For n = 12:
- 12 = 2^2 * 3 = prime(1+0)^2 * prime(1+1),
- f(12) = 2 + x,
- f(12)^2 = 4 + 4*x + x^2,
- a(12) = prime(1+0)^4 * prime(1+1)^4 * prime(1+2) = 2^4 * 3^4 * 5 = 6480.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A001222(n))

Crossrefs

Cf. A000040, A001221, A001222, A031368, A055396, A061395, A296857, A297845.

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]-1)^(f[i,2]*f[j,2])))

Formula

For any n > 0 and k > 0:

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

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

- A055396(a(n)) = 2*A055396(n)-1 + [n=1],

- A061395(a(n)) = 2*A061395(n)-1 + [n=1],

- a(A000040(n)) = A031368(n),

- a(A000040(n)^k) = A031368(n)^(k^2).