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.

A003369 Numbers that are the sum of 2 positive 7th powers.

Original entry on oeis.org

2, 129, 256, 2188, 2315, 4374, 16385, 16512, 18571, 32768, 78126, 78253, 80312, 94509, 156250, 279937, 280064, 282123, 296320, 358061, 559872, 823544, 823671, 825730, 839927, 901668, 1103479, 1647086, 2097153, 2097280, 2099339, 2113536
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
3909794986386 is in the sequence as 3909794986386 = 57^7 + 57^7.
6061605477062 is in the sequence as 6061605477062 = 19^7 + 67^7.
26019535290982 is in the sequence as 26019535290982 = 61^7 + 81^7. (End)

Links

Crossrefs

Cf. A000404 (2 squares), A003325 (2 cubes), A003336 (2 4th), A003347 (2 5th), A003358 (2 6th), A088719 (2 distinct 7th), A003380 (2 8th).
Cf. A001015 (seventh powers).

Programs

  • Maple
    N:= 10^7: # to get all terms <= N
S:= select(`<=`, {seq(seq(a^7+b^7, a=1..b), b=1..floor(N^(1/7)))}, N):
sort(convert(S, list)); # Robert Israel, Sep 03 2017
  • Mathematica
    lst={}; Do[If[(a^7+b^7)==n, AppendTo[lst, n]], {n, 200000}, {a, (n/2)^(1/7)}, {b, a, (n-a^7)^(1/7)}]; lst (* XU Pingya, Sep 03 2017 *)
Module[{upto=10},Select[Union[Total/@Tuples[Range[upto]^7,2]],#<= (upto^7)&]] (* Harvey P. Dale, Feb 04 2019 *)

A088677 Numbers that can be represented as j^6 + k^6, with 0 < j < k, in exactly one way.

Original entry on oeis.org

65, 730, 793, 4097, 4160, 4825, 15626, 15689, 16354, 19721, 46657, 46720, 47385, 50752, 62281, 117650, 117713, 118378, 121745, 133274, 164305, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 531442, 531505, 532170, 535537, 547066
Offset: 1

Views

Author

Cino Hilliard, Nov 22 2003

Keywords

Comments

Conjecture: no number can be expressed as such a sum in more than one way.
Ekl (1996) has searched and found no solutions to the 6.2.2 Diophantine equation A^6 + B^6 = C^6 + D^6 with sums less than 7.25 * 10^26. - Jonathan Vos Post, May 04 2006

Examples

			65 = 1^6 + 2^6.

Links

Crossrefs

Cf. A003358, A088719 (7th powers).

Programs

  • Mathematica
    lst={};e=6;Do[Do[x=a^e;Do[y=b^e;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/e)],a+1,-1}],{a,Floor[n^(1/e)],1,-1}],{n,2*8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
  • PARI
    powers2(m1,m2,p1) = { for(k=m1,m2, a=powers(k,p1); if(a==1,print1(k",")) ); } powers(n,p) = { z1=0; z2=0; c=0; cr = floor(n^(1/p)+1); for(x=1,cr, for(y=x+1,cr, z1=x^p+y^p; if(z1 == n,c++); ); ); return(c) }

Extensions

Edited by Don Reble, May 03 2006

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

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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