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 1-10 of 10 results.

A003336 Numbers that are the sum of 2 positive 4th powers.

Original entry on oeis.org

2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8192, 8962, 10001, 10016, 10081, 10256, 10625
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers k such that k = x^4 + y^4 has a solution in positive integers x, y.
There are no squares in this sequence. - Altug Alkan, Apr 08 2016
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
16378801 is in the sequence as 16378801 = 43^4 + 60^4.
39126977 is in the sequence as 39126977 = 49^4 + 76^4.
71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)

Links

Crossrefs

5906 is the first term in A060387 but not in this sequence. Cf. A020897.
Cf. A088687 (2 distinct 4th powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
Cf. A000583 (4th powers).

Programs

  • Mathematica
    nn=12; Select[Union[Plus@@@(Tuples[Range[nn],{2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)
Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 08 2016 *)
  • PARI
    list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1,4), for(y=1, min(sqrtnint(lim-x^4,4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024
  • PARI
    T=thueinit('x^4+1,1);
is(n)=#thue(T,n)>0 && !issquare(n) \\ Charles R Greathouse IV, Feb 26 2017
  • Python
    def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  return sorted(p2)
print(aupto(10625)) # Michael S. Branicky, Mar 18 2021

Formula

{i: A216284(i) > 0}. - R. J. Mathar, Jun 04 2021

A088703 Numbers of form x^5 + y^5, x,y > 0 and x <> y.

Original entry on oeis.org

33, 244, 275, 1025, 1056, 1267, 3126, 3157, 3368, 4149, 7777, 7808, 8019, 8800, 10901, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 32800, 33011, 33792, 35893, 40544, 49575, 59050, 59081, 59292, 60073, 62174, 66825, 75856, 91817
Offset: 1

Views

Author

Cino Hilliard, Nov 22 2003

Keywords

Comments

Up to n = 100000, no instances occur where n is the sum of two distinct 5th powers in two different ways. Conjecture: no number can be expressed as the sum of two 5th powers in more than one way: A046881.

Examples

			33 = 2^5 + 1^5, so 33 is in sequence. 64 = 2^5 + 2^5 is not.

References

  • Guy, Richard K., Unsolved Problems in Number Theory, 2nd Ed., Springer-Verlag(1994), pp. 140.

Links

Crossrefs

Subsequence of A003347.
Cf. A088687 (4th powers), A088677 (6th powers), A046881 (bounds for double reps).

Programs

  • Mathematica
    lst={};e=5;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,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Union[#[[1]]^5+#[[2]]^5&/@Subsets[Range[10],{2}]] (* Harvey P. Dale, Apr 25 2012 *)
  • 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 Ralf Stephan, Dec 30 2004

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

A230562 Smallest number that is the sum of 2 positive 4th powers in >= n ways.

Original entry on oeis.org

0, 2, 635318657
Offset: 0

Views

Author

Jonathan Sondow, Oct 25 2013

Keywords

Comments

Hardy and Wright say that a(3) is unknown.
Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums."

Examples

			0 = (empty sum).
2 = 1^4 + 1^4.
635318657 = 59^4 + 158^4 = 133^4 + 134^4.

References

  • R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, 2008; section 21.11.

Links

Crossrefs

Cf. A003824, A016078, A018786, A046881, A088687, A088848, A088849, A088867, A230561.

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

A133137 a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of 4th powers of 2 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73
Offset: 1

Views

Author

Jonathan Vos Post, Sep 21 2007

Keywords

Comments

This is to A000583 as A031980 is to A000578. First differs from complement of A088687 at 83522 = 1^4 + 17^4.

Crossrefs

Cf. A000583, A031980, A088687.

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

A088727 a(n) is the (n-1)st smallest number that is the sum of 2 distinct positive n-th powers.

Original entry on oeis.org

5, 28, 97, 1025, 4160, 18571, 390626, 1953637, 9824674, 53022429, 2176782337, 13060702208, 78368947065, 471258726400, 2973697798081, 232630513987208, 1628413598172593, 11398896347634610, 79793365809239777, 559022701241487132, 4041442752425255185
Offset: 2

Views

Author

Cino Hilliard, Nov 23 2003

Keywords

Examples

			Sums of 2 distinct k-th powers:
k=2:   5,   10,   13,    17,    20,    25,     26, ...
k=3:   9,   28,   35,    65,    72,    91,    126, ...
k=4:  17,   82,   97,   257,   272,   337,    626, ...
k=5:  33,  244,  275,  1025,  1056,  1267,   3126, ...
k=6:  65,  730,  793,  4097,  4160,  4825,  15626, ...
k=7: 129, 2188, 2315, 16385, 16512, 18571,  78126, ...
k=8: 257, 6562, 6817, 65537, 65792, 72097, 390626, ...
[edited by _Jon E. Schoenfield_, Mar 07 2018]

Crossrefs

Cf. A004431, A024670, A088687. - Michel Marcus, Mar 07 2018

Extensions

Edited Name and more terms from Jon E. Schoenfield, Mar 04 2018

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
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