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A088867
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Non-squarefree elements of A003824, i.e., primitive solutions to a^4 + b^4 = c^4 + d^4 with non-squarefree value on both sides of the equation.
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1
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680914892583617, 55683917506335026, 2056314197022256097, 3267700501872475297, 4544031582110882417, 10555434261160919777, 12361929340136667457, 23076050051029379057, 335875812638910622082
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OFFSET
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1,1
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COMMENTS
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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.
Among the first 516 terms of A003824, there are 31 non-squarefree 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
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LINKS
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Table of n, a(n) for n=1..9.
D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4
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FORMULA
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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, ...
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EXAMPLE
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a(1) = A003824(18) = 680914892583617 = 17^2*89*61657*429361 is the first non-squarefree term of A003824. - M. F. Hasler, Mar 05 2012
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PROG
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(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", ")) ) }
(PARI) select(A003824, t->!issquarefree(t)) \\ - M. F. Hasler, Mar 05 2012
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CROSSREFS
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Cf. A003824, A088848, A088849.
Sequence in context: A086438 A104873 A172585 * A159042 A129935 A173405
Adjacent sequences: A088864 A088865 A088866 * A088868 A088869 A088870
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Nov 26 2003
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STATUS
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approved
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