A353008 a(n) is the smallest positive k such that k^2 + 1 has 2*n divisors, or -1 if no such k exists.

1, 3, 7, 13, 182, 43, 1068, 47, 268, 443, 15905182, 157, 1832311432, 14557, 16432, 307, 255250280182, 1407, 355101282318, 3307, 92682, 3626068, 21346690797155182, 993, 313932, 120813568, 51982, 16693, 982692130687379186432, 2943, 2444574943897581751068, 2163

Offset: 1

Author

Jon E. Schoenfield, May 15 2022

Keywords

• nonn

Comments

From Jon E. Schoenfield, Jun 14 2024: (Start)

For integers k, neither 3 nor 4 ever divides k^2 + 1, so there exists no prime p < 5 such that p^2 divides k^2 + 1.

For n <= 32, the only n for which the 5-adic valuation of a(n)^2 + 1 is not gpf(n) - 1 is n = 16 (see Examples).

Conjecture: a(n) is never -1. (End)

Examples

			From _Jon E. Schoenfield_, Jun 14 2024: (Start)
From a(5) = 182 because 182 is the smallest positive integer k such that k^2 + 1 has 2*5 divisors: 182^2 + 1 = 33125 = 5^4 * 53.
a(16) = 307 because 307 is the smallest positive integer k such that k^2 + 1 has 2*16 divisors: 307^2 + 1 = 94250 = 2 * 5^3 * 377.
a(31) = 2444574943897581751068: 2444574943897581751068^2 + 1 = 5975946656331864965715445578098297119140625 = 5^30 * 6416623862896477837609. (End)
		

Crossrefs

Cf. A006530, A112765, A193432, A347193.

Extensions

a(26), a(29), and a(31) corrected by Jon E. Schoenfield, Jun 14 2024