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A266927 Perfect powers of the form x^2 + y^2 where x and y are positive integers. 5
8, 25, 32, 100, 125, 128, 169, 225, 289, 400, 512, 625, 676, 841, 900, 1000, 1156, 1225, 1369, 1521, 1600, 1681, 2025, 2048, 2197, 2500, 2601, 2704, 2809, 3025, 3125, 3364, 3600, 3721, 4225, 4624, 4900, 4913, 5329, 5476, 5625, 5832, 6084, 6400, 6724, 7225, 7569 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Intersection of A000404 and A001597.

A134422 is a subsequence.

Obviously, this sequence contains all numbers of the form 2^(2*n+1), for n > 0.

Motivation for this sequence is the equation m^k = x^2 + y^2 where m,x,y > 0, k >= 2. - Altug Alkan, Jan 11 2016

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

25 is a term because 25 = 5^2 = 3^2 + 4^2.

32 is a term because 32 = 2^5 = 4^2 + 4^2.

125 is a term because 125 = 5^3 = 10^2 + 5^2.

169 is a term because 169 = 13^2 = 5^2 + 12^2.

MAPLE

N:= 10000: # to get all terms <= N

g:= proc(k)

    local F, F1, F2, F3, f;

    F:= ifactors(k)[2];

    F2, F:= selectremove(f->f[1]=2, F);

    F1, F3:= selectremove(f -> f[1] mod 4 = 1, F);

    if F1 <> [] then

       if hastype(map(f -> f[2], F3), odd) then

          seq(k^j, j=2..floor(log[k](N)), 2)

       else seq(k^j, j=2..floor(log[k](N)))

       fi

    elif F2 = [] or F2[1][2]::even or hastype(map(f -> f[2], F3), odd) then NULL

    else seq(k^j, j=3..floor(log[k](N)), 2)

    fi

end proc:

sort(convert(map(g, {$2..floor(sqrt(N))}), list)); # Robert Israel, Jan 11 2016

MATHEMATICA

lim = 7600; fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Union@ Flatten@ Table[a^2 + b^2, {a, Floor[Sqrt[lim - 1]]}, {b, a, Floor[Sqrt[lim - a^2]]}], fQ] (* Michael De Vlieger, Jan 06 2016, after N. J. A. Sloane and J. H. Conway at A000404 and Ant King at A001597 *)

PROG

(PARI) is(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}

for(n=1, 1e4, if((ispower(n) || n==1) && is(n), print1(n, ", ")));

CROSSREFS

Cf. A000404, A001597, A004171, A134422.

Sequence in context: A045860 A081504 A030796 * A240591 A116086 A270739

Adjacent sequences:  A266924 A266925 A266926 * A266928 A266929 A266930

KEYWORD

nonn

AUTHOR

Altug Alkan, Jan 06 2016

STATUS

approved

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Last modified September 18 17:02 EDT 2020. Contains 337170 sequences. (Running on oeis4.)