A088867 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.

680914892583617, 55683917506335026, 2056314197022256097, 3267700501872475297, 4544031582110882417, 10555434261160919777, 12361929340136667457, 23076050051029379057, 335875812638910622082

Offset: 1

Author

Cino Hilliard, Nov 26 2003

Keywords

• nonn

Comments

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 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

Examples

			a(1) = A003824(18) = 680914892583617 = 17^2*89*61657*429361 is the first nonsquarefree term of A003824. - _M. F. Hasler_, Mar 05 2012
		

Links

• D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4

Crossrefs

Cf. A003824, A088848, A088849.

Programs

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

Formula

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, ...