A267702 -id:A267702 - cp's OEIS frontend

A267686 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.

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

Offset: 1

Altug Alkan, Jan 19 2016

nonn

Inspired by intersection of A000408, A000583 and A003325.
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.

			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.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408, A000583, A003325, A267702.

isA000408(n) = {my(a, b); a=1; while(a^2+1A003325(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, ", ")));

Added missing term a(32), Chai Wah Wu, Jan 31 2016

A272174 Values of a^3 + b^3 such that the equation a^3 + b^3 = x^2 + y^2 + z^2 is not soluble where a, b > 0 and x, y, z >= 0.

Original entry on oeis.org

28, 351, 407, 559, 855, 1008, 1343, 1792, 2071, 3087, 3383, 3439, 3591, 3887, 4375, 4439, 5103, 5488, 6119, 6175, 7471, 8343, 9207, 10864, 10991, 11375, 11772, 12175, 12231, 12383, 12636, 12679, 13167, 13895, 14023, 14167, 14364, 14911, 16263, 16956, 17199, 17919, 17999

Offset: 1

Altug Alkan, Apr 21 2016

Intersection of A003325 and A004215.

			28 is a term because 28 = 1^3 + 3^3 and 28 is the sum of 4 but no fewer nonzero squares.

Cf. A003325, A004215, A267702.

isA004215(n) = {local(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0);}
isA003325(n) = {for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));}
lista(nn) = for(n=1, nn, if(isA004215(n) && isA003325(n), print1(n, ", ")));

A273843 Numbers that are the average of 3 nonzero squares and the average of 2 positive cubes.

Original entry on oeis.org

1, 8, 14, 27, 36, 63, 64, 76, 112, 140, 172, 185, 216, 234, 260, 288, 343, 364, 378, 427, 504, 512, 536, 608, 666, 679, 728, 729, 868, 896, 972, 1000, 1030, 1099, 1112, 1120, 1161, 1270, 1331, 1376, 1404, 1463, 1480, 1628, 1688, 1728, 1750, 1764, 1859, 2052, 2080, 2156

Offset: 1

Altug Alkan, Jun 01 2016

Values of (x^3 + y^3)/2 such that (x^3 + y^3)/2 = (a^2 + b^2 + c^2)/3 where x, y, a, b, c > 0, is soluble.

			14 is a term because 14 = (1^3 + 3^3)/2 = (1^2 + 4^2 + 5^2)/3.

Cf. A000408, A003325, A267702.

repQ[n_,k_,e_] := {} != Quiet@ IntegerPartitions[n, {k}, Range[n^ (1/e) ]^e, 1]; Select[Range@ 2156, repQ[2*#,2,3] && repQ[3*#,3,2] &] (* Giovanni Resta, Jun 03 2016 *)

isA000408(n) = my(a, b) ; a=1 ; while(a^2+1A003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
lista(nn) = for(n=1, nn, if(isA003325(2*n) && isA000408(3*n), print1(n, ", ")));

cp's OEIS Frontend

A267686 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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A272174 Values of a^3 + b^3 such that the equation a^3 + b^3 = x^2 + y^2 + z^2 is not soluble where a, b > 0 and x, y, z >= 0.

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A273843 Numbers that are the average of 3 nonzero squares and the average of 2 positive cubes.

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