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.

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A068537 Numbers which can be written as the sum of 2 like powers (x^n + y^n; n>1 & x,y>0).

Original entry on oeis.org

2, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 28, 29, 32, 33, 34, 35, 37, 40, 41, 45, 50, 52, 53, 54, 58, 61, 64, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 91, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 126, 128, 129, 130, 133, 136, 137, 145, 146
Offset: 1

Views

Author

Shawn Schafer (coolrelish(AT)hotmail.com), Mar 22 2002

Keywords

Examples

			2 = 1^2 + 1^2; 5 = 1^2 + 2^2; 8 = 2^2 + 2^2; 9 = 1^3 + 2^3; 10 = 1^2 + 3^2; 13 = 2^2 + 3^2; 16 = 2^3 + 2^3; 17 = 1^2 + 4^2; ..... 33 = 1^5 + 2^5; etc...

Crossrefs

Superset of A000404, A003325, A003336, A003347, A003358, A003369, A003380, A003391.

Extensions

More terms from Michel Marcus, Aug 07 2013
More terms from Sean A. Irvine, Feb 21 2024

A085304 Least number of 4th powers required to represent n!.

Original entry on oeis.org

1, 1, 2, 6, 9, 10, 15, 15, 9, 10, 15, 6, 12, 12
Offset: 0

Views

Author

Labos Elemer, Jun 30 2003

Keywords

Examples

			n=6: 6!=720=625+81+14,length-of-solution=16>=a(6)
but 6!=720=2.256+13.16 seems shortest solution a(6)=15
after, see also A046046
n=7: 7!=5040=3.1296+4.256+8.16 so a(7)<=15 (uncertain);
n=8: a(8)<=9 because 8!=4.10000+1.256+4.16.

Links

Crossrefs

Cf. A000142, A084355, A060387, A003336-A003346, A046046, A046044-A046050.

Formula

"Shortest" solutions to n!=Sum[x(j)^4], j=1, .., m[n] with minimal value of m[n]: a(n)=Min{m[n]}. Per analogiam A084355.

Extensions

a(7)-a(11) from John W. Layman, Aug 13 2004
a(12) from Sean A. Irvine, Feb 11 2010
a(13) from Sean A. Irvine, Feb 15 2010

A291826 Numbers k such that k^5 is sum of 2 nonzero 6th powers.

Original entry on oeis.org

32, 2048, 23328, 131072, 500000, 1492992, 3764768, 8388608, 17006112, 32000000, 56689952, 95551488, 154457888, 240945152, 364500000, 536870912, 772402208, 1088391168, 1160290625, 1505468192, 2048000000, 2744515872, 3628156928, 4737148448, 6115295232
Offset: 1

Views

Author

XU Pingya, Sep 03 2017

Keywords

Comments

If a^6 + b^6 = m, then (m^4*a)^6 + (m^4*b)^6 = m^25 = (m^5)^5 is 5th power. Therefore A003358(n)^5 is a term of this sequence for all n.
When k in this sequence, k*(n^6) (n >= 2) is also in this sequence.
If h = (i^6)*(j^6 + 1)^5 for (i >= 1 and j >= 1), then h is in this sequence. It appears that this equation generates all terms of the sequence. - Kieran Bhaskara, Aug 03 2019

Examples

			32^5 = 16^6 + 16^6, so 32 is in the sequence.
1160290625^5 = 17850625^6 + 35701250^6, so 1160290625 is in the sequence.

Crossrefs

Cf. A000404, A004431, A003325, A050801, A003336, A003347, A003358.

Programs

  • Mathematica
    lst={};Do[If[IntegerQ[(n^5-a^6)^(1/6)],AppendTo[lst,n]],{n,7*10^9},{a,(n^5/2)^(1/6)}]; lst

A291849 Numbers k such that k^3 is the sum of two nonzero 4th powers.

Original entry on oeis.org

8, 128, 648, 2048, 4913, 5000, 10368, 19208, 32768, 52488, 78608, 80000, 117128, 165888, 228488, 307328, 397953, 405000, 524288, 551368, 668168, 839808, 912673, 1042568, 1257728, 1280000, 1555848, 1874048, 2238728, 2654208, 3070625, 3125000, 3655808, 4251528
Offset: 1

Views

Author

XU Pingya, Sep 04 2017

Keywords

Comments

If a^4 + b^4 = m, then (m^2 * a)^4 + (m^2 * b)^4 = m^9 = (m^3)^3 is a cube. Therefore A003336(n)^3 are terms of this sequence.
When k is in this sequence, k(n^4), for n > 1, is also in this sequence.

Examples

			8^3 = 2^9 = 4^4 + 4^4, so 8 is in the sequence.
4913^3 = 17^9 = 17^8 * (1 + 2^4) = 289^4 + 578^4, so 4913 is in the sequence.

Crossrefs

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

Programs

  • Mathematica
    fourthPowerFlags = Union@Flatten@Table[a^4 + b^4 && GCD[a, b] == 1, {a, 4}, {b, a, 4}]; Take[Union@Flatten@Table[k^4 * fourthPowerFlags[[j]]^3, {k, 27}, {j, 6}], 34]

A291850 Numbers k such that k^2 is the sum of two positive 5th powers.

Original entry on oeis.org

8, 256, 1944, 6655, 8192, 25000, 35937, 62208, 134456, 212960, 262144, 344605, 453962, 472392, 692759, 800000, 1149984, 1288408, 1617165, 1990656, 2970344, 4302592, 6075000, 6814720, 8388608, 8732691, 11358856, 14526784, 15116544, 19808792, 20796875, 22168288
Offset: 1

Views

Author

XU Pingya, Sep 04 2017

Keywords

Comments

If a^5 + b^5 = m, then (ma)^5 + (mb)^5 = m^6 = (m^3)^2 is square. Therefore A003347(n)^3 are terms of this sequence.
When k is in this sequence, k * (n^5) (n = 2, 3, ... ) is also in this sequence.

Examples

			8^2 = 2^5 + 2^5, so 8 is in the sequence.
6655^2 = 22^5 + 33^5, so 6655 is in the sequence.

Crossrefs

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

Programs

  • Mathematica
    lst={};Do[If[IntegerQ[(n^2-a^5)^(1/5)],AppendTo[lst,n]],{n,9000},{a,(n^2/2)^(1/5)}]; lst
  • PARI
    upto(n) = {
    my(res = List(), u = n^2, i5);
    for(i = 1, sqrtnint(u, 5),
        i5 = i^5;
        for(j = i, sqrtnint(u - i5, 5),
            c = i5 + j^5;
            if(issquare(c, &sc),
                listput(res, sc))));
    Set(res)} \\ David A. Corneth, Jun 17 2025

A291851 Numbers k such that k^3 is the sum of two positive 5th powers.

Original entry on oeis.org

4, 128, 972, 1089, 4096, 12500, 31104, 34848, 59536, 67228, 75625, 131072, 236196, 264627, 400000, 644204, 995328, 1050625, 1115136, 1485172, 1605289, 1905152, 2151296, 2420000, 3037500, 3403125, 4194304, 5679428, 7558272, 8468064, 9771876, 9904396, 9966649
Offset: 1

Views

Author

XU Pingya, Sep 04 2017

Keywords

Comments

If a^5 + b^5 = m, then (ma)^5 + (mb)^5 = m^6 = (m^2)^3 is a cube. Therefore the square of each term of A003347 is a term of this sequence.
When k is in this sequence, k * (n^5), for n > 1, is also in this sequence.

Examples

			4^3 = 2^6 = 2^5 + 2^5, so 4 is in the sequence.
1089^3 = 33^5 + 66^5, so 1089 is in the sequence.

Crossrefs

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

Programs

  • Mathematica
    lst={};Do[If[IntegerQ[(n^3-a^5)^(1/5)],AppendTo[lst,n]],{n,10^7},{a,(n^3/2)^(1/5)}]; lst
Previous Showing 71-76 of 76 results.

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