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.

Previous Showing 11-16 of 16 results.

A179509 Numbers that can be written uniquely as sum of a square and a nonnegative cube.

Original entry on oeis.org

0, 2, 4, 5, 8, 10, 12, 16, 24, 25, 26, 27, 28, 31, 33, 37, 43, 44, 49, 50, 52, 57, 63, 68, 72, 73, 76, 80, 81, 82, 91, 101, 113, 121, 122, 125, 126, 127, 128, 134, 141, 144, 148, 150, 152, 161, 164, 169, 170, 171, 174, 177, 185, 189, 197, 204, 206, 208, 216, 217, 220
Offset: 1

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Author

Reinhard Zumkeller, Jul 17 2010

Keywords

Comments

A045634(a(n)) = 1;
subsequence of A022549.

Links

A123291 Numbers that are sum of a square and a nonnegative cube (with repetition).

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 8, 9, 9, 10, 12, 16, 17, 17, 24, 25, 26, 27, 28, 31, 33, 36, 36, 37, 43, 44, 49, 50, 52, 57, 63, 64, 64, 65, 65, 68, 72, 73, 76, 80, 81, 82, 89, 89, 91, 100, 100, 101, 108, 108, 113, 121, 122, 125, 126, 127, 128, 129, 129, 134, 141, 144, 145, 145, 148
Offset: 1

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Author

Zak Seidov, Oct 11 2006

Keywords

Comments

Cf. A022549 Sum of a square and a nonnegative cube (without repetition), A022550 Numbers that are not the sum of a square and a nonnegative cube.

Examples

			Each of 1, 9, 17, 36 appear two times because
1=0^2+1^3=1^2+0^3, 9=1^2+2^3=3^2+0^3, 17=3^2+2^3==4^2+1^3, 36=3^2+3^3==6^2+0^3;
225 appears three times because 225=3^2+6^3=10^2+5^3=15^2+0^3;
1025 appears four times because 1025=5^2+10^3=30^2+5^3=31^2+4^3=32^2+1^3, etc.

Crossrefs

Cf. A022549, A022550.

Programs

  • Mathematica
    Lim=148; s=Ceiling[Sqrt[Lim]];c=Ceiling[Lim^(1/3)];sq=Range[0,s]^2;cb=Range[0,c]^3;seq={};Do[AppendTo[seq,sq[[i]]+cb[[j]]],{i,s+1},{j,c+1}];Sort[Select[seq,#<=Lim&]] (* James C. McMahon, Nov 19 2024 *)

A169618 Table with T(n,k) = the number of ways to represent k as the sum of a square and a cube modulo n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 6, 6, 2, 2, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 11, 8, 12, 2, 6, 3, 12, 20, 4, 4, 12, 4, 4, 4, 15, 15, 6, 6, 6, 6, 6, 6, 15, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 18, 18, 6, 6, 18, 18, 6, 6, 18, 18, 6, 6, 13, 11, 18, 8, 20, 15, 6
Offset: 1

Views

Author

Franklin T. Adams-Watters, Dec 03 2009

Keywords

Comments

The top left corner is T(1,0).
It appears that this table does not contain any 0's.
It appears that row n is constant iff n is squarefree, and no prime divisor of n is == 1 (mod 6). It is not hard to show that such rows are constant, since the cubes are equi-distributed in such moduli.

Examples

			The 6 ways to represent 0 (mod 4) are 0^2+0^3, 0^2+2^3, 1^2+3^3, 2^2+0^3, 2^2+2^3, and 3^2+3^3.

Crossrefs

Cf. A022549, A002476, A045309.

Programs

  • PARI
    al(n)=local(v);v=vector(n);for(i=0,n-1,for(j=0,n-1,v[(i^2+j^3)%n+1]++));v

A273553 Numbers n such that n! is the sum of a square and a nonnegative cube.

Original entry on oeis.org

0, 1, 2, 4, 19, 21
Offset: 1

Views

Author

Altug Alkan, May 25 2016

Keywords

Comments

Numbers n such that n! = x^2 + y^3 where x, y are integers and y >= 0, is soluble.
a(7) > 33. - Giovanni Resta, Jun 03 2016

Examples

			4 is a term because 4! = 4^2 + 2^3.
19 is a term because 19! = 345427200^2 + 132480^3.
21 is a term because 21! = 6389296200^2 + 2173500^3.

Crossrefs

Cf. A000142, A022549.

A367984 Numbers that are the sum of a nonnegative cube and a fourth power of a nonnegative integer.

Original entry on oeis.org

0, 1, 2, 8, 9, 16, 17, 24, 27, 28, 43, 64, 65, 80, 81, 82, 89, 108, 125, 126, 141, 145, 206, 216, 217, 232, 256, 257, 264, 283, 297, 320, 343, 344, 359, 381, 424, 472, 512, 513, 528, 593, 599, 625, 626, 633, 652, 689, 729, 730, 745, 750, 768, 810, 841, 968, 985, 1000
Offset: 1

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Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000578, A000583, A022549, A100291.

Programs

  • Maple
    M:= 10000: # for terms <= M
Cubes:= {seq(i^3, i=0..floor(M^(1/3)))}:
Fourths:= {seq(i^4, i=0..floor(M^(1/4)))}:
sort(convert(select(`<=`,{seq(seq(a+b,a=Cubes),b=Fourths)},M),list)); # Robert Israel, Dec 05 2024

A214922 Numbers of the form x^2 + y^2 + z^3 + w^3 (x, y, z, w >= 0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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
Offset: 1

Views

Author

Philippe Deléham, Jul 20 2012

Keywords

Comments

Conjecture: 23 is the only number not in this sequence.
Not the same as A004830: 239 is a term of this sequence but not of A004830. - R. J. Mathar and Joerg Arndt, Jul 28 2012
There are no other missing numbers from 24 to 10^8. - Giovanni Resta, Oct 12 2019

Examples

			22 = 2^2 + 4^2 + 1^3 + 1^3, 22 is in this sequence.

Crossrefs

Cf. A001481, A004999, A022549, A022550, A022551, A022552.
Previous Showing 11-16 of 16 results.

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