cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A051386 Numbers whose 4th power is the sum of two positive cubes.

Original entry on oeis.org

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 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
Offset: 1

Views

Author

Jud McCranie

Keywords

Comments

n such that n^4 = r^3 + s^3 has a solution with r>0, s>0.
By multiplying n^4 = r^3 + s^3 by n^3, also numbers whose 7th power is expressible as the sum of positive cubes.
When n is the sum of 2 positive cubes (A003325) there is a trivial solution: e.g., 133 is a term in A003325, 133=2^3+5^3 and 133^4=(2*133)^3+(5*133)^3. - Zak Seidov, Oct 17 2011
From Robert Israel, Jun 01 2015: (Start)
Slightly more generally, if x^3 + y^3 = u*v^4, then (u*v*w^3)^4 = (u*w^4*x)^3 + (u*w^4*y)^3, so u*v*w^3 is in the sequence for any w >= 1.
There are at least five pairs of adjacent numbers in the sequence: (133,134),(182,183), (854,855), (1842,1843), (3473,3474). Are there infinitely many?
(End)

Examples

			134^4 = 469^3 + 603^3.

Links

Crossrefs

Cf. A003325, A051387.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
Cubes:= {seq(x^3,x=1..floor(N^(4/3)))}:
select(n -> nops(map(t -> n^4-t, Cubes) intersect Cubes)>0, [$1..N]); # Robert Israel, Jun 01 2015

A051388 Numbers whose 4th power can be expressed as the sum of two positive cubes in more than one way.

Original entry on oeis.org

1729, 2457, 4104, 4914, 4977, 8001, 8216, 10773, 13832, 15561, 16263, 19656, 20683, 32832, 39312, 39816, 40033, 46683, 64008, 64232, 65728, 66339, 80236, 86184, 110656, 110808, 124336, 124488, 127062, 130104, 132678, 132867, 134379, 149389, 157248, 165464, 166887, 171288
Offset: 1

Views

Author

Jud McCranie

Keywords

Examples

			1729^4 = 1729^3 + 20748^3 = 15561^3 + 17290^3.

Links

Crossrefs

Cf. A051386, A051387.
Cf. A226777, A226950.

Programs

  • PARI
    for(i=1, 10^5, x=i^4; c=0; for(j=1, floor(sqrtn(x/2,3)), if(ispower(x-j^3,3), c++)); if(c>1, print(i" "c))) \\ Donovan Johnson, Aug 03 2009

Extensions

More terms from John W. Layman, Feb 24 2003
a(18)-a(24) from Donovan Johnson, Aug 03 2009
a(25)-a(38) from Donovan Johnson, Dec 03 2010

A282872 Numbers in A003325 whose 4th power is the sum of two positive cubes in a nontrivial way.

Original entry on oeis.org

2457, 4914, 4977, 8001, 8216, 10773, 15561, 16263, 19656, 39816, 64008, 66339, 80236, 86184, 124336, 124488, 127062, 130104, 132678, 132867, 157248, 166887, 201717, 221832, 238329, 252035, 290871, 307125, 318528, 338821, 358036, 406952, 411021, 420147, 421876
Offset: 1

Views

Author

Chai Wah Wu, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A001235, A003325, A051387, A282873.

Formula

A003325 INTERSECT A051387.

A282873 Taxicab numbers (A001235) whose 4th power is the sum of two positive cubes in a nontrivial way.

Original entry on oeis.org

10202696, 29791125, 48137544, 70957971, 81621568, 238329000, 275472792, 288975141, 385100352, 387352719, 553514689, 567663768, 652972544, 692612137, 728274456, 1051871977, 1104726168, 1275337000, 1299713688, 1402390152, 1484204904, 1906632000, 2203782336, 2311801128
Offset: 1

Views

Author

Chai Wah Wu, Feb 24 2017

Keywords

Comments

If n = a^3 + b^3, then n^4 has a trivial decomposition as a sum of 2 cubes: n^4 = (an)^3 + (bn)^3.

Links

Crossrefs

Cf. A001235, A003325, A051387, A282872.

Formula

A001235 INTERSECT A051387.

A273615 Numbers k such that k^4 is the average of two positive cubes while k is not.

Original entry on oeis.org

329, 518, 566, 662, 732, 741, 777, 804, 806, 876, 921, 998, 1029, 1092, 1236, 1238, 1317, 1497, 1526, 1596, 1812, 1862, 1929, 1988, 2181, 2316, 2604, 2632, 2757, 4204, 4396, 4446, 4684, 5068, 5548, 5782, 5838, 5856, 5928, 5982, 6124, 6126, 6216
Offset: 1

Views

Author

Altug Alkan, May 26 2016

Keywords

Comments

If k is the average of two positive cubes, then k^4 is also the average of two positive cubes. So this sequence focuses on the solutions that are not trivial.

Examples

			329 is a term because 329 is not the average of two positive cubes while 329^4 = (1833^3 + 2585^3)/2.

Links

Crossrefs

Cf. A003325, A051387.

Programs

  • Maple
    Q:=  proc(x) local t;
  for t in select(t -> t^3<=x and 4*t^3 > x and x/t - t^2 mod 3 = 0,
        numtheory:-divisors(x)) do
    if issqr((x/t - t^2)/3)  then return true fi
  od:
  false
end proc:
select(x -> not(Q(x)) and Q(x^4), [$1..10000]); # Robert Israel, May 26 2016
  • Mathematica
    Q[x_] := Module[{s, t}, s = Select[Divisors[x], #^3 <= x && 4*#^3 > x && Mod[x/# - #^2, 3] == 0 &]; For[t = 1, t <= Length[s], t++, If[IntegerQ@Sqrt[(x/s[[t]] - s[[t]]^2)/3],  Return[True]]]; False];
Reap[For[x = 1, x <= 10000, x++, If[!Q[x] && Q[x^4], Print[x]; Sow[x]]]][[2, 1]] (* Jean-François Alcover, May 18 2023, after Robert Israel *)
  • PARI
    isA003325(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^4) && !isA003325(2*n), print1(n, ", ")));
  • PARI
    T=thueinit('z^3+1);
isA003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
is(n)=isA003325(2*n^4) && !isA003325(2*n) \\ Charles R Greathouse IV, May 27 2016
Showing 1-5 of 5 results.

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