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
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
A098970
Numbers k such that (12*k)^2 can be expressed as the sum of the cubes of two distinct primes.
Original entry on oeis.org
19, 67695, 411292, 1134035, 1184876, 2112836, 2455255, 4073384, 11293009, 16171470, 18589912, 34388501, 63609329, 63711615, 117446600, 166530856, 284034387, 449805631, 637548135, 685361103, 783484793, 888180400, 1121365940
Offset: 1
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
A103255
Integers x > 0 such that x^3 + y^3 = z^2 for some y > 0, z > 0, and gcd(x,y) = 1.
Original entry on oeis.org
1, 2, 11, 23, 37, 56, 57, 65, 112, 122, 193, 217, 242, 305, 312, 433, 592, 781, 851, 877, 889, 913, 1001, 1064, 1177, 1201, 1346, 1376, 1617, 1633, 1706, 1729, 1801, 1953, 1960, 1969, 2137, 2162, 2184, 2257, 2345, 2480, 2543, 2920, 3071, 3081, 3482, 3641, 3889, 4019
Offset: 1
x=11, y=37, 11^3 + 37^3 = 228^2. 11 is the third entry in the list.
The pairs [x,y] = [a(n),a(?)] for the first few terms are [1, 2], [2, 1], [11, 37], [23, 1177], [37, 11], [56, 65], [57, 112], [65, 56], [112, 57], [122, 1201], [193, 3482], [217, 312], [242, 433]. [_Joerg Arndt_, Sep 30 2012]
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[ k : k in [1..100] | exists{P : P in IntegralPoints(EllipticCurve([0,k^3])) | P[1] gt 0 and P[2] ne 0 and GCD(Integers()!P[1],k) eq 1} ]; // Geoff Bailey
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(* This program uses z-values from A099426 b-file. To get 50 terms, the first 200 z-values suffice, the result being the same with the whole b-file of 300 z-values. *)
terms = 50;
zz = Import["https://oeis.org/A099426/b099426.txt", "Table"][[1 ;; 4 terms, 2]];
r[z_] := {x, y, z} /. ToRules[Reduce[GCD[x, y] == 1 && 0Jean-François Alcover, Jun 13 2019 *)
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is_A103255(x, lim)=
{ /* Warning: just how big lim has to be is unclear */
my(x3=x^3);
for (y=1, lim,
if ( gcd(x,y) != 1, next() );
if ( issquare(x3+y^3), return(1) );
);
return(0);
}
/* Using lim=10^6 reproduces all terms <= 1000: */
for (n=1,1000, if( is_A103255(n, 10^6), print1(n,", ")) );
/* Joerg Arndt, Sep 30 2012 */
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# apparently inefficient as of version 5.2
def is_A103255(n):
E = EllipticCurve([0, n^3])
E.gens(descent_second_limit=16);
for p in E.integral_points():
if p[0] > 0 and p[1] > 0 and gcd(p[1], n) == 1:
return true
return false
[n for n in (1..60) if is_A103255(n)]
# Peter Luschny, Sep 29 2012
Recomputed and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007
A251781
Numbers whose square is the sum of two distinct positive cubes.
Original entry on oeis.org
3, 24, 81, 98, 168, 192, 228, 312, 375, 525, 588, 648, 671, 784, 847, 1014, 1029, 1183, 1225, 1261, 1323, 1344, 1536, 1824, 2187, 2496, 2646, 2888, 3000, 3993, 4200, 4225, 4536, 4563, 4644, 4704, 5184, 5368, 6156, 6272, 6292, 6371, 6591, 6696, 6776, 6877, 8112
Offset: 1
3^2 = 1^3 + 2^3; 24^2 = 4^3 + 8^3.
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def aupto(limit):
c = [i**3 for i in range(1, int(limit**(2/3))+2) if i**3 <= limit**2]
cc = [c1 + c2 for i, c1 in enumerate(c) for c2 in c[i+1:]]
return sorted([i for i in range(1, limit+1) if i*i in cc])
print(aupto(8122)) # Michael S. Branicky, Mar 24 2021
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L = []
for k in range(1,10^3):
for l in range(k + 1,10^3):
if is_square(k**3+l**3):
L.append(sqrt(k**3+l**3))
A183150
Semiprimes s such that s^2 is expressible as the sum of two positive cubes.
Original entry on oeis.org
4, 671, 1261, 6371, 127499, 377567, 897623, 1984009, 4266107, 4870741, 4974061, 5491823, 24923137, 26784757, 28192247, 33601933, 36295069, 44091347, 44988481, 61717319, 95327051, 97587433, 99712367, 142798573, 149982097, 193405967
Offset: 1
a(1) = 4 = 2*2 because 4^2 = 16 = 2^3 + 2^3 . a(2) = 671 = 11 * 61 and 56^3 + 65^3 = 671^2 = 450241. a(3) = 1261 = 13 * 97 and 1261^2 = 57^3 + 112^3. a(6) = 897623 = 107 * 8389.
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Select[Range[194*10^6],PrimeOmega[#]==2&&Length[ PowersRepresentations[ #^2,2,3]]>0&] (* The program takes a long time to run. *) (* Harvey P. Dale, Feb 27 2016 *)
Showing 1-7 of 7 results.
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