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 1-7 of 7 results.

A003347 Numbers that are the sum of 2 positive 5th powers.

Original entry on oeis.org

2, 33, 64, 244, 275, 486, 1025, 1056, 1267, 2048, 3126, 3157, 3368, 4149, 6250, 7777, 7808, 8019, 8800, 10901, 15552, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 32800, 33011, 33614, 33792, 35893, 40544, 49575, 59050, 59081, 59292, 60073, 62174, 65536, 66825, 75856
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
917552689 is in the sequence as 917552689 = 17^5 + 62^5.
2557575000 is in the sequence as 2557575000 = 45^5 + 75^5.
5828050944 is in the sequence as 5828050944 = 56^5 + 88^5. (End)

Links

Crossrefs

Cf. A000584, A003336, A003348, A003358, A088703.

Programs

A155468 Numbers that are sums of 8th powers of 2 distinct positive integers.

Original entry on oeis.org

257, 6562, 6817, 65537, 65792, 72097, 390626, 390881, 397186, 456161, 1679617, 1679872, 1686177, 1745152, 2070241, 5764802, 5765057, 5771362, 5830337, 6155426, 7444417, 16777217, 16777472, 16783777, 16842752, 17167841, 18456832, 22542017, 43046722, 43046977, 43053282
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

Keywords

Examples

			1^8 + 2^8 = 257, 1^8 + 3^8 = 6562, 2^8 + 3^8 = 6817, ...

Links

Crossrefs

Cf. A003380, A088719 (distinct 7th), A088677 (distinct 6th), A088703, A088687, A024670 (distinct 3rd), A004431 (distinct 2nd).

Programs

  • Mathematica
    lst={};e=8;Do[Do[x=a^e;Do[y=b^e;If[x+y==n,Print[n,",",Date[]];AppendTo[lst,n]],{b,Floor[(n-x)^(1/e)],a+1,-1}],{a,Floor[n^(1/e)],1,-1}],{n,4*8!}];lst
  • PARI
    list(lim)=my(v=List(),t); lim\=1; for(m=2,sqrtnint(lim-1,8), t=m^8; for(n=1,min(sqrtnint(lim-t,8),m-1), listput(v,t+n^8))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

Extensions

8 more terms. - R. J. Mathar, Sep 07 2017
More terms from Chai Wah Wu, Nov 05 2017

A008515 5-dimensional centered cube numbers.

Original entry on oeis.org

1, 33, 275, 1267, 4149, 10901, 24583, 49575, 91817, 159049, 261051, 409883, 620125, 909117, 1297199, 1807951, 2468433, 3309425, 4365667, 5676099, 7284101, 9237733, 11589975, 14398967, 17728249, 21647001, 26230283, 31559275, 37721517, 44811149, 52929151, 62183583, 72689825
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are never prime, as a(n) = (2n+1)*(n^4+2*n^3+4*n^2+3*n+1). - Jonathan Vos Post, Aug 18 2011
zeta(5) = 1 / (a(1) - 1^10 / (a(2) - 2^10 / (a(3) - 3^10 / ... ))) [From Tito Piezas III mathoverflow question 265688 comment]. - Michael Somos, Jul 06 2017

Links

Crossrefs

Apart from the first term, a subsequence of A088703.

Programs

  • GAP
    List([0..40], n-> n^5+(n+1)^5); # G. C. Greubel, Nov 09 2019
  • Magma
    [n^5+(n+1)^5: n in [0..40]]; // Bruno Berselli, Aug 25 2011
  • Maple
    seq(n^5+(n+1)^5, n=0..40);
  • Mathematica
    Sum[(Range[40]+j-2)^5, {j,2}] (* G. C. Greubel, Nov 09 2019 *)
  • PARI
    a(n) = n^5+(n+1)^5;
  • Sage
    [n^5+(n+1)^5 for n in (0..40)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = n^5 + (n+1)^5 = 2*n^5 +5*n^4 +10*n^3 +10*n^2 +5*n +1.
From Bruno Berselli, Aug 25 2011: (Start)
G.f.: (1+x)*(1 +26*x +66*x^2 +26*x^3 +x^4)/(1-x)^6.
a(n) = -a(-n-1).
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). (End)
E.g.f.: (1 +32*x +105*x^2 +90*x^3 +25*x^4 +2*x^5)*exp(x). - G. C. Greubel, Nov 09 2019

A155469 Numbers that are the sum of 2 (not-distinct) numbers; nonzero square and cube, including repetitions.

Original entry on oeis.org

2, 5, 9, 10, 12, 17, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, 65, 68, 72, 73, 76, 80, 82, 89, 89, 91, 100, 101, 108, 108, 113, 122, 126, 127, 128, 129, 129, 134, 141, 145, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

Keywords

Comments

5=2^2+1^3, 12=2^2+2^3, 17=3^2+2^3, 31=2^2+3^3, 43=4^2+3^3, 65=1^2+4^3, 65=8^2+1^3, 100=6^2+4^3, ...

Crossrefs

Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468

Programs

  • Mathematica
    lst={};Do[Do[Do[a=x^2+y^3;If[a>n,Break[]];If[a==n,AppendTo[lst,n]],{y,5!}],{x,5!}],{n,4*5!}];lst

A155470 Numbers that are the sum of 2 numbers; nonzero square and cube, including repetitions, squareNumber <> cubeNumber.

Original entry on oeis.org

5, 9, 10, 17, 17, 24, 26, 28, 31, 33, 37, 43, 44, 50, 52, 57, 63, 65, 65, 68, 72, 73, 76, 82, 89, 89, 91, 100, 101, 108, 108, 113, 122, 126, 127, 128, 129, 129, 134, 141, 145, 145, 148, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204, 206, 208, 217, 220
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

Keywords

Comments

17=3^2+2^3, 17=4^2+1^3, 31=2^2+3^3, 43=4^2+3^3, 65=1^2+4^3, 65=8^2+1^3, 100=6^2+4^3, ...

Crossrefs

Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468, A155469

Programs

  • Mathematica
    lst={};Do[Do[Do[If[x!=y,a=x^2+y^3;If[a>n,Break[]];If[a==n,AppendTo[lst,n]]],{y,5!}],{x,5!}],{n,4*5!}];lst

A155472 Numbers that are the sum of 2 (not-distinct) numbers; nonzero power3 and power5, including repetitions.

Original entry on oeis.org

2, 9, 28, 33, 40, 59, 65, 96, 126, 157, 217, 244, 248, 251, 270, 307, 344, 368, 375, 459, 513, 544, 586, 730, 755, 761, 972, 1001, 1025, 1032, 1032, 1051, 1088, 1149, 1240, 1243, 1332, 1363, 1367, 1536, 1574, 1729, 1753, 1760, 1971, 2024, 2198, 2229, 2355
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

Keywords

Comments

40=2^3+2^5, 1032=2^3+4^5 = 1032=10^3+2^5, 1971=12^3+3^5, ...

Crossrefs

Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468, A155469, A155470

Programs

  • Mathematica
    lst={};Do[Do[Do[a=x^3+y^5;If[a>n,Break[]];If[a==n,AppendTo[lst,n]],{y,5!}],{x,5!}],{n,7!}];lst

A155473 Numbers of the form x^3+y^5, with x,y>0 and x<>y.

Original entry on oeis.org

9, 28, 33, 59, 65, 96, 126, 157, 217, 244, 248, 251, 307, 344, 368, 375, 459, 513, 544, 586, 730, 755, 761, 972, 1001, 1025, 1032, 1032, 1051, 1149, 1240, 1243, 1332, 1363, 1367, 1536, 1574, 1729, 1753, 1760, 1971, 2024, 2198, 2229, 2355, 2440, 2745, 2752
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

Keywords

Comments

Numbers with more than one of these representations are repeated for each of them.
This concerns 1032 = 2^3+4^5 = 10^3+2^5 or 9504 = 12^3+6^5 = 21^3+3^5, for example (see A035046).

Examples

			59=3^3+2^5, 157=5^3+2^5, 513=8^3+1^5, 586=7^3+3^5, ...

Crossrefs

Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468, A155469, A155470, A155472

Programs

  • Mathematica
    lst={};Do[Do[Do[If[x!=y,a=x^3+y^5;If[a>n,Break[]];If[a==n,AppendTo[lst,n]]],{y,5!}],{x,5!}],{n,7!}];lst

Extensions

Edited by R. J. Mathar, Mar 02 2009
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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