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%I #37 Sep 08 2022 08:44:58
%S 1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,
%T 28,29,30,31,33,34,35,36,37,39,40,42,44,45,46,47,48,49,50,51,52,53,54,
%U 55,56,58,59,60,61,62,63,64,65,66,67,69,71,72,73,74,75,76,77,78,79,80
%N Numbers k such that k^2+1 is squarefree.
%C Estermann proved that a(n) ~ kn with k = 1.117...; more precisely, there are cx + O(x^(2/3) log x) terms up to x, where c = 1/k = Product (1 - 2/p^2) where the product is over primes p which are 1 mod 4. Heath-Brown improves the error term to O(x^(7/12) log x). - _Charles R Greathouse IV_, Oct 16 2017, corrected by _Amiram Eldar_, Jul 08 2020
%C There are 89489 terms up to 10^5, 894856 terms up to 10^6, 8948417 up to 10^7, 89484102 up to 10^8, and 894841314 up to 10^9. - _Charles R Greathouse IV_, Nov 26 2017, corrected and extended by _Amiram Eldar_, Jul 08 2020
%H Michael De Vlieger, <a href="/A049533/b049533.txt">Table of n, a(n) for n = 1..10000</a>
%H T. Estermann, <a href="https://eudml.org/doc/159528">Einige Sätze über quadratfreie Zahlen</a>, Math. Ann. 105 (1931), pp. 653-662.
%H D. R. Heath-Brown, <a href="https://arxiv.org/abs/1010.6217">Square-free values of n^2 + 1</a>, Acta Arithmetica 155:1 (2012), pp. 1-13; arXiv:1010.6217 [math.NT], 2010-2012.
%F Numbers k such that A059592(k) = 1. - _Reinhard Zumkeller_, Nov 08 2006
%e 10 is a member because 10^2 + 1 = 100 + 1 = 101 is squarefree.
%e Reasons why certain numbers are excluded: 7^2+1 = 2*5^2, 18^2+1 = 13*5^2, 32^2+1 = 41*5^2, 38^2+1 = 5*17^2, 41^2+1 = 2*29^2, 43^2+1 = 74*5^2, 57^2+1 = 130*5^2, 82^2+1 = 269*5^2. - Neven Juric, Oct 06 2008
%t Select[Range@ 80, SquareFreeQ[#^2 + 1] &] (* _Michael De Vlieger_, Aug 09 2017 *)
%o (Magma) [ n: n in [1..100] | IsSquarefree(n^2+1) ]; // _Vincenzo Librandi_, Dec 25 2010
%o (PARI) isok(n) = issquarefree(n^2+1); \\ _Michel Marcus_, Feb 09 2016
%Y Complement of A049532.
%Y Cf. A059592, A069987, A335963.
%K nonn
%O 1,2
%A _Labos Elemer_
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