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

%I A353008 #63 Jun 15 2024 22:50:46
%S A353008 1,3,7,13,182,43,1068,47,268,443,15905182,157,1832311432,14557,16432,
%T A353008 307,255250280182,1407,355101282318,3307,92682,3626068,
%U A353008 21346690797155182,993,313932,120813568,51982,16693,982692130687379186432,2943,2444574943897581751068,2163
%N A353008 a(n) is the smallest positive k such that k^2 + 1 has 2*n divisors, or -1 if no such k exists.
%C A353008 From _Jon E. Schoenfield_, Jun 14 2024: (Start)
%C A353008 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.
%C A353008 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).
%C A353008 Conjecture: a(n) is never -1. (End)
%e A353008 From _Jon E. Schoenfield_, Jun 14 2024: (Start)
%e A353008 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.
%e A353008 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.
%e A353008 a(31) = 2444574943897581751068: 2444574943897581751068^2 + 1 = 5975946656331864965715445578098297119140625 = 5^30 * 6416623862896477837609. (End)
%Y A353008 Cf. A006530, A112765, A193432, A347193.
%K A353008 nonn
%O A353008 1,2
%A A353008 _Jon E. Schoenfield_, May 15 2022
%E A353008 a(26), a(29), and a(31) corrected by _Jon E. Schoenfield_, Jun 14 2024