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A267686
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Positive integers n such that n^4 = a^3 + b^3 = x^2 + y^2 + z^2 where x, y, z, a and b are positive integers, is soluble.
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1
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9, 28, 35, 54, 65, 72, 91, 126, 133, 134, 152, 182, 183, 189, 201, 217, 219, 224, 243, 250, 273, 278, 280, 309, 341, 344, 351, 370, 399, 407, 422, 432, 453, 468, 497, 513, 520, 539, 559, 576, 579, 637, 651, 658, 686, 728, 730, 737, 756, 793, 854, 855
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OFFSET
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1,1
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COMMENTS
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Corresponding fourth powers are 6561, 614656, 1500625, 8503056, 17850625, 26873856, 68574961, 252047376, 312900721, 322417936, 533794816, 1097199376, 1121513121, 1275989841, 1632240801, 2217373921, 2300257521, 2517630976, 3486784401, ...
2 is the first number that its 4th power, 2^4, is the sum of 2 positive cubes and is not the sum of 3 nonzero squares. 16 is the second number for this case. So 2 and 16 are not in this sequence.
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LINKS
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EXAMPLE
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9 is a term because 9^4 = 9^3 + 18^3 = 1^2 + 28^2 + 76^2.
28 is a term because 28^4 = 28^3 + 84^3 = 64^2 + 144^2 + 768^2.
35 is a term because 35^4 = 70^3 + 105^3 = 1^2 + 600^2 + 1068^2.
54 is a term because 54^4 = 162^3 + 162^3 = 12^2 + 264^2 + 2904^2.
399 is a term because 399^4 = 665^3 + 2926^3 = 17^2 + 11236^2 + 158804^2.
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PROG
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(PARI) isA000408(n) = {my(a, b); a=1; while(a^2+1<n, b=1; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1)); b++; ); a++; ); return(0); }
T=thueinit('z^3+1);
isA003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
for(n=3, 1e3, if(isA000408(n^4) && isA003325(n^4), print1(n, ", ")));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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