A099532
Lesser of a,b where n^2 = a^3 + b^3; a,b > 0 and gcd(a,b)=1. The greater of a,b is the corresponding term in A099533 and n, which is used to order this sequence, is the corresponding term in A099426.
Original entry on oeis.org
1, 11, 56, 57, 217, 242, 305, 851, 23, 122, 592, 1706, 1376, 1001, 2257, 1960, 193, 781, 889, 1729, 3071, 2920, 5896, 1346, 2137, 2184, 1633, 8313, 7034, 1953, 7379, 9097, 6104, 10712, 4097, 12369, 7082, 12265, 13441, 15709, 18314, 5833, 8792, 8929, 3641
Offset: 1
a(2) is 11 because term A099426(2) is 228, 228^2 = 11^3 + 37^3 and 11 < 37.
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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(x", ")))) \\ Charles R Greathouse IV, Nov 06 2014
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 */
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/* 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
A282639
Greater value of a coprime pair (x,y) satisfying x^3+y^3=z^2.
Original entry on oeis.org
2, 37, 65, 112, 312, 433, 877, 1064, 1177, 1201, 1617, 1969, 2345, 2480, 2543, 3081, 3482, 4019, 4440, 5160, 5985, 6337, 6489, 7729, 8663, 8848, 9265, 10274, 10753, 11821, 12071, 12688, 13937, 14345, 15520, 15962, 16296, 16352, 17761, 21923, 22753, 26208
Offset: 1
(x,y,z) = (11,37,228), (56,65,671), (57,112,1261), (217,312,6371), (242,433,9765), (851,877,35928), (305,1064,35113), (23,1177,40380), (122,1201,41643), (592,1617,66599), (1706,1969,112245), (1376,2345,124501), (1001,2480,127499), (2257,2543,167160), (1960,3081,191771)... - _R. J. Mathar_, Jan 22 2025
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A282639 := proc(n)
option remember ;
local x,y ;
if n =1 then
2
else
for y from procname(n-1)+1 do
for x from 1 to y do
if igcd(x,y) = 1 then
if issqr(x^3+y^3) then
return y;
end if;
end if;
end do:
end do:
end if;
end proc:
seq(A282639(n),n=1..17) ; # R. J. Mathar, Jan 22 2025
A282095
Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.
Original entry on oeis.org
11, 46, 52, 117, 142, 198, 236, 286, 415, 488, 524, 549, 621, 666, 835, 873, 908, 970, 1001, 1199, 1388, 1432, 1692, 1757, 1962, 1964, 1971, 2035, 2041, 2366, 2392, 2630, 2655, 2681, 2702, 2815, 2826, 3195, 3421, 3544, 3664, 3715, 4048, 4070, 4097, 4356
Offset: 1
2^2 + 11^2 = 5^3, so 11 is in the sequence.
9^2 + 46^2 = 13^3, so 46 is in the sequence.
47^2 + 52^2 = 17^3, so 52 is in the sequence.
44^2 + 117^2 = 25^2, so 117 is in the sequence.
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# slow version for demonstration only.
isA282095 := proc(y)
local x,z3 ;
for x from 1 to y do
if igcd(x,y) = 1 then
z3 := x^2+y^2 ;
if isA000578(z3) then
return true ;
end if;
end if;
end do:
return false ;
end proc:
for y from 1 do
if isA282095(y) then
printf("%d,\n",y) ;
end if;
end do:
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okQ[y_] := Module[{x, z3}, For[x=1, xJean-François Alcover, Dec 04 2017, after R. J. Mathar *)
Showing 1-4 of 4 results.
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