A099426
Numbers n where n^2 = x^3 + y^3; x,y>0 and gcd(x,y)=1.
Original entry on oeis.org
3, 228, 671, 1261, 6371, 9765, 35113, 35928, 40380, 41643, 66599, 112245, 124501, 127499, 167160, 191771, 205485, 255720, 297037, 377567, 532392, 546013, 647569, 681285, 812340, 897623, 1043469, 1125683, 1261491, 1431793, 1433040, 1584828, 1783067, 1984009, 2107391, 2372903, 2440893, 2484469, 2548557
Offset: 1
228 is in the sequence because 228^2 = 11^3 + 37^3 and gcd(11, 37) = 1.
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n = 10^7; n2 = n^2; x = 1; x3 = x^3; Reap[ While[x3 < n2, y = x + 1; y3 = y^3; While[y3 < n2, If[GCD[x, y] == 1, s = x3 + y3; If[ IntegerQ[r = Sqrt[s]], Print[r]; Sow[r]; Break[]]]; y += 1; y3 = y^3]; x += 1; x3 = x^3]][[2, 1]] // Sort (* Jean-François Alcover, Jan 11 2013, translated from Joerg Arndt's 2nd Pari program *)
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is_A099426(n)=
{
my(n2=n^2, k=1, k3=1, r);
while( k3 < n2,
if ( ispower(n2-k3, 3, &r),
if ( gcd(r,k)==1, return(1) );
);
k+=1; k3=k^3;
);
return(0);
}
for (n=1,10^8, if( is_A099426(n), print1(n,", ")) );
/* Joerg Arndt, Sep 30 2012 */
-
/* compute all terms below a threshold at once, terms need to be sorted */
{ N = 10^7; N2 = N^2;
x=1; x3=x^3;
while ( x3 < N2,
y=x+1; y3=y^3;
while ( y3 < N2,
if ( gcd(x,y) == 1,
s = x3 + y3;
if ( issquare(s, &r), print(r); break(); );
);
y+=1; y3 = y^3;
);
x+=1; x3 = x^3;
);}
/* Joerg Arndt, Sep 30 2012 */
-
for(s=2,1e5,for(x=1,s\2,my(y=s-x);if(gcd(x,y)>1,next); if(issquare(x^3+y^3), print1(s", ")))) \\ Charles R Greathouse IV, Nov 06 2014
A099533
Greater of a,b where n^2 = a^3 + b^3; a,b>0 and gcd(a,b)=1. The lesser of a,b is the corresponding term in A099532 and n, which is used to order this sequence, is the corresponding term in A099426.
Original entry on oeis.org
2, 37, 65, 112, 312, 433, 1064, 877, 1177, 1201, 1617, 1969, 2345, 2480, 2543, 3081, 3482, 4019, 4440, 5160, 6337, 6489, 5985, 7729, 8663, 9265, 10274, 8848, 10753, 12688, 11821, 12071, 14345, 13937, 16352, 15520, 17761, 16296, 15962, 21923
Offset: 1
Term #2 is 37 because term #2 of A099426 is 228, 228^2 = 11^3 + 37^3 and 37 > 11.
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for(s=2,1e5,for(x=1,s\2,my(y=s-x);if(gcd(x,y)>1,next); if(issquare(x^3+y^3), print1(y", ")))) \\ Charles R Greathouse IV, Nov 06 2014
A320662
Numbers k for which there are numbers 0 < m <= k such that k^3 + m^3 is a square.
Original entry on oeis.org
2, 8, 18, 21, 26, 32, 37, 46, 50, 65, 70, 72, 84, 88, 91, 98, 104, 105, 112, 128, 148, 162, 184, 189, 190, 200, 234, 242, 249, 260, 273, 280, 288, 312, 330, 333, 336, 338, 345, 352, 354, 364, 371, 392, 407, 414, 416, 420
Offset: 1
8^3 + 4^3 = 512 + 64 = 576 = 24^2, so 8 is part of the sequence.
18^3 + 9^3 = 5832 + 729 = 6561 = 81^2, so 18 is part of the sequence.
91^3 + 65^3 = 753571 + 274625 = 1028196 = 1014^2, so 91 is part of the sequence.
7^3 + 0^3 = 343 + 0 = 343, 7^3 + 1^3 = 343 + 1 = 344, 7^3 + 2^3 = 343 + 8 = 351,7^3 + 4^3 = 343 + 64 = 407, 7^3 + 5^3 = 343 + 125 = 468, 7^3 + 6^3 = 343 + 216 = 559 and 7^3 + 7^3 = 343 + 343 = 686. Numbers 343, 344, 351, 407, 468, 559 and 686 are not squares, so 7 is not part of the sequence.
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A320662 := proc(n)
option remember ;
local m,k ;
if n =1 then
2
else
for k from procname(n-1)+1 do
for m from 1 to k do
if issqr(k^3+m^3) then
return k ;
end if;
end do:
end do:
end if;
end proc:
seq(A320662(n),n=1..40) ; # R. J. Mathar, Jan 22 2025
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Select[Range@ 420, AnyTrue[Range[#1]^3 + #2, IntegerQ@ Sqrt@ # &] & @@ {#, #^3} &] (* Michael De Vlieger, Nov 05 2018 *)
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is(n) = for(m=1, n, if(issquare(n^3+m^3), return(1))); 0 \\ Felix Fröhlich, Oct 22 2018
Showing 1-3 of 3 results.
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