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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).
3
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
1,2
COMMENTS
This sequence is the main diagonal of A297845.
This sequence has similarities with A296857.
LINKS
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A001222(n))
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).
EXAMPLE
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.
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]-1)^(f[i, 2]*f[j, 2])))
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Dec 30 2017
STATUS
approved