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

1,1

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

Links

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

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, ...

Example

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

Prog

(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

Crossrefs

Cf. A003824, A088848, A088849.

Sequence in context: A336968 A321709 A338608 * A256234 A298821 A159042

Adjacent sequences: A088864 A088865 A088866 * A088868 A088869 A088870

Keyword

nonn

Author

Cino Hilliard, Nov 26 2003

Status

approved