A316780 a(n) is the least positive integer k such that ceiling(sqrt(A046315(n)*k))^2 - A046315(n)*k is a square.

1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 7, 1, 7, 3, 9, 3, 3, 1, 9, 9, 3, 11, 1, 5, 5, 13, 3, 1, 15, 15, 5, 1, 17, 3, 5, 1, 17, 7, 3, 17, 1, 7, 19, 1, 21, 3, 5, 7, 23, 5, 1, 25, 9, 1, 5, 25, 9, 27, 3, 27, 1, 29, 5, 11, 29, 3, 11, 1, 11, 5, 3, 33, 1, 35, 13

Offset: 1

Author

Arnauld Chevallier, Jul 13 2018

Keywords

• nonn

Comments

Fermat's factorization helper multiplier for the n-th odd semiprime.

a(n) is the least positive integer such that A046315(n) * a(n) can be factorized with a single iteration of Fermat's factorization method. Using the factorization of a(n), we can then deduce the prime factors of A046315(n). Example for n = 35490: A046315(n) = 199163 and a(n) = 40; ceiling(sqrt(199163*40)) = 2823; 199163*40 = 2823^2 - 2809 = 2823^2 - 53^2 = (2823+53)(2823-53) = 2876*2770, leading to 199163*(2*2*2*5) = (2*2*719)*(2*5*277) and eventually 199163 = 719*277.

Examples

			a(18) = 7 because the 18th odd semiprime is A046315(18) = 93, ceiling(sqrt(93*7))^2 - 93*7 = 25 is a perfect square and 7 is the least positive integer for which this holds.
		

Crossrefs

Cf. A046315.