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%I #38 Feb 26 2024 01:22:03
%S 3,5,11,13,21,29,43,59,61,67,69,77,83,85,93,115,123,131,133,139,141,
%T 155,157,165,173,187,203,205,211,213,219,221,227,229,237,259,267,277,
%U 283,285,291,309,317,331,347,355,357,365,371,373,381,389,403,411,419,421
%N Odd numbers c such that c*(c^2 - 1)/4 is squarefree.
%C Original title was: "Numbers c such that (c^2-1)c is square free and gcd(c-1,c,c+1)=1", but (c^2-1)c is never squarefree for odd c, and gcd(n,n+1) is always = 1. - _M. F. Hasler_, Nov 03 2013
%C These numbers c with distribution a+b=c such that a=(c-1)/2 (see A172186) and b=(c+1)/2 (see A179019) have minimal possible values with function L(a,b,c) = log(c)/log(N(a,b,c)) = log(c)/log((c^2-1)c/4).
%C This function is minimal orbital in hypothesis (a,b,c).
%C There are no numbers or distributions which have value L less than log(c)/log((c^2-1)*c/4).
%C Equivalently, odd squarefree numbers c such that (c^2 - 1)/4 is also squarefree. - _Jon E. Schoenfield_, Feb 13 2023
%C The asymptotic density of this sequence is Product_{p prime} (1 - 3/p^2) = A206256 = 0.125486980905... (Tsang, 1985). - _Amiram Eldar_, Feb 26 2024
%H Charles R Greathouse IV, <a href="/A179017/b179017.txt">Table of n, a(n) for n = 1..10000</a>
%H Kai-Man Tsang, <a href="https://doi.org/10.1112/S0025579300011049">The distribution of r-tuples of square-free numbers</a>, Mathematika, Vol. 32, No. 2 (1985), pp. 265-275.
%F a(n) = 2*A172186(n) + 1. - _Bernard Schott_, Mar 06 2023
%t aa = {}; Do[If[(GCD[x, (x - 1)/2] == 1) && (GCD[x, (x + 1)/2] == 1) && (GCD[(x - 1)/2, (x + 1)/2] == 1), If[SquareFreeQ[(x^2 - 1) x/4], AppendTo[aa, x]]], {x, 2, 1000}]; aa
%o (PARI) forstep(n=3,421,2,issquarefree(n*(n^2-1)/4)&&print1(n",")) \\ _M. F. Hasler_, Nov 03 2013
%o (PARI) is(n)=n%2 && issquarefree(n) && issquarefree(n^2\4) \\ _Charles R Greathouse IV_, Mar 11 2014
%Y Cf. A172120, A172121, A147298, A147299, A147300, A147301, A147302, A147306, A147307, A147638, A147639, A147640, A147641, A147642, A147643, A143700, A172186, A179019, A206256.
%K nonn
%O 1,1
%A _Artur Jasinski_, Jun 24 2010
%E Edited by _M. F. Hasler_, Nov 03 2013
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