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A304435
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Numbers n such that n^5 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).
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3
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3, 5, 6, 8, 10, 13, 17, 19, 20, 24, 25, 26, 28, 29, 34, 36, 37, 40, 41, 45, 50, 52, 53, 54, 58, 61, 62, 65, 68, 73, 74, 75, 80, 81, 82, 85, 88, 89, 90, 96, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 130, 136, 137, 145, 146, 148, 149, 150, 153, 157, 160, 164, 168, 169, 170, 173, 176, 178, 180, 181, 185, 192, 193, 194, 197
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OFFSET
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1,1
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COMMENTS
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Motivated by the search of solutions to a^n + b^(2n+2)/4 = (perfect square), which arises when searching solutions to x^n + y^(n+1) = z^(n+2) of the form x = a*z, y = b*z. It turns out that many solutions are of the form a^n = d (b^(n+1) + d), where d is a perfect power.
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LINKS
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EXAMPLE
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3^5 = 3^3 + 6^3; 5^5 = 10^2 + 55^2, 6^5 = 2^5 + 88^2, ...
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MAPLE
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LIM:= 200^5: P:={seq(seq(x^k, k=3..floor(log[x](LIM))), x=2..floor(LIM^(1/3)))}:
is_A304435:= proc(n) local n5, Pp; n5:= n^5; if remove(t -> subs(t, x)<=1 or subs(t, y)<=1 or subs(t, x-y)=0, [isolve(x^2+y^2=n^5)]) <> [] then return true fi; Pp:= map(t ->n5-t, P minus {n5, n5/2}); (Pp intersect P <> {}) or (select(issqr, Pp) <> {})
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MATHEMATICA
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M = 200; LIM = M^5;
P = Flatten @ Table[Table[x^k, {k, 3, Floor[Log[x, LIM]]}], {x, 2, Floor[ LIM^(1/3)]}];
filterQ[n_] := Module[{n5 = n^5, Pp, x, y}, If[Solve[x > 1 && y > 1 && x != y && x^2 + y^2 == n5, {x, y}, Integers] != {}, Return[True]]; Pp = n5 - (P ~Complement~ {n5, n5/2}); (Pp ~Intersection~ P) != {} || Select[Pp, IntegerQ[Sqrt[#]]&] != {}];
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PROG
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(PARI) N=200; L=N^5; P=List(); for(x=2, sqrtnint(L, 3), for(k=3, logint(L, x), listput(P, x^k))); P=Set(P);
is_A304435(n)={for(i=1, #s=sum2sqr(n=n^5), vecmin(s[i])>1 && s[i][1]!=s[i][2] && return(1)); for(i=1, #P, n>P[i]||return; ispower(n-P[i])&& P[i]*2 != n && return(1))} \\ For sum2sqr() see A133388.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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