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%I #18 Aug 09 2015 15:21:15
%S 680914892583617,55683917506335026,2056314197022256097,
%T 3267700501872475297,4544031582110882417,10555434261160919777,
%U 12361929340136667457,23076050051029379057,335875812638910622082
%N Nonsquarefree elements of A003824, i.e., primitive solutions to a^4 + b^4 = c^4 + d^4 with nonsquarefree value on both sides of the equation.
%C Original definition was: Numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways that have at least one repeated factor.
%C Among the first 516 terms of A003824, there are 31 nonsquarefree terms. None of these are expressible in more than 2 ways as sum of two 4th powers. However, some of them, as 4544031582110882417, 12361929340136667457, 335875812638910622082, ..., have gcd(a,b) > 1, for one of the decompositions a^4 + b^4. - _M. F. Hasler_, Mar 05 2012
%H D. J. Bernstein, <a href="http://cr.yp.to/sortedsums/two4.1000000">List of 516 primitive solutions p^4 + q^4 = r^4 + s^4</a>
%F omega(n)<>bigomega(n) for n = a^4+b^4 = c^4+d^4 for distinct a, b, c, d. n=635318657, 3262811042, ..., 680914892583617, ..., 962608047985759418078417, ...
%e a(1) = A003824(18) = 680914892583617 = 17^2*89*61657*429361 is the first nonsquarefree term of A003824. - _M. F. Hasler_, Mar 05 2012
%o (PARI) \ begin a new session and type \r x4data.txt (evaluated Bernstein data) This will allow using %1 as the initial value. omegax4py42(n) = { for (i = 1, n, x = eval( Str("%", i) ); y=omega(x); y1 =bigomega(x); if(y<>y1,print1(x",")) ) }
%o (PARI) select(A003824, t->!issquarefree(t)) \\ _M. F. Hasler_, Mar 05 2012
%Y Cf. A003824, A088848, A088849.
%K nonn
%O 1,1
%A _Cino Hilliard_, Nov 26 2003
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