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 1-4 of 4 results.

A095809 Least positive number having exactly n partitions into three squares.

Original entry on oeis.org

1, 9, 41, 81, 146, 194, 306, 369, 425, 594, 689, 866, 1109, 1161, 1154, 1361, 1634, 1781, 1889, 2141, 2729, 2609, 3626, 3366, 3566, 3449, 3506, 4241, 4289, 4826, 5066, 5381, 7034, 5561, 6254, 7229, 7829, 8186, 8069, 8126, 8609, 8921, 8774, 10386
Offset: 1

Views

Author

T. D. Noe, Jun 07 2004

Keywords

Comments

Note that a square can be zero.

Examples

			41 is the least number having three partitions: 41 = 0+16+25 = 1+4+36 = 9+16+16.

Links

Crossrefs

Apart from initial term, same as A000437.
Cf. A094739 (n having a unique partition into three squares), A095811 (greatest number having exactly n partitions into three squares), A124970.

Programs

  • Mathematica
    lim=200; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n

A115288 a(n) is the smallest number representable in exactly n ways as a sum of 2 triangular numbers and one square (each of them >= 0).

Original entry on oeis.org

0, 1, 4, 7, 10, 16, 22, 25, 64, 46, 70, 67, 92, 85, 160, 115, 106, 136, 200, 157, 190, 172, 256, 235, 568, 277, 370, 337, 400, 367, 340, 550, 556, 442, 1102, 445, 472, 631, 610, 535, 682, 697, 652, 1075, 956, 850, 1984, 865, 1172, 997, 862, 1081, 1462, 1135, 1060
Offset: 1

Views

Author

Giovanni Resta, Jan 19 2006

Keywords

Examples

			a(4)=7 since 7 can be expressed in 4 ways, 7= T(3)+T(1)+0^2 = T(3)+T(0)+1^2 = T(2)+T(0)+2^2 = T(2)+T(2)+1^2 and none of the numbers from 0 to 6 can be expressed in 4 ways.

Crossrefs

Cf. A115289, A000437, A061262, A330861.

Programs

  • Maple
    a := [seq(0,n=0..100)] ;
for k from 0 do
    a330861 := A330861(k) ;
    if a330861 <= nops(a) then
        if op(a330861,a) = 0 then
            a := subsop(a330861=k,a) ;
            print(a) ;
        end if;
    end if;
    if not member(0,[op(2..nops(a),a)]) then
        break;
    end if;
end do: # R. J. Mathar, Apr 28 2020
  • Mathematica
    V=Table[0, {i, 2500}]; T[n]:=n(n+1)/2; Do[a=T[i]+T[j]+k^2;If[a<2500, V[[a+1]]++ ], {i, 0, 71}, {j, 0, i}, {k, 0, 50}]; Table[Position[V, z][[1, 1]]-1, {z, 60}]

A124970 Smallest positive integer which can be expressed as the ordered sum of 3 squares in exactly n different ways.

Original entry on oeis.org

7, 1, 9, 41, 81, 146, 194, 306, 369, 425, 594, 689, 866, 1109, 1161, 1154, 1361, 1634, 1781, 1889, 2141, 2729, 2609, 3626, 3366, 3566, 3449, 3506, 4241, 4289, 4826, 5066, 5381, 7034, 5561, 6254, 7229, 7829, 8186, 8069, 8126, 8609, 8921, 8774, 10386, 11574, 11129
Offset: 0

Views

Author

Artur Jasinski, Nov 14 2006, Nov 20 2006

Keywords

Links

Crossrefs

Cf. A000451, A095809, A000437, A122699, A124966-A124971.

Programs

  • Mathematica
    f[n_] := Block[{k = 1}, While[Length@PowersRepresentations[k, 3, 2] != n, k++]; k]; Table[f[n], {n, 0, 44}] (* Ray Chandler, Oct 31 2019 *)
  • Python
    from collections import Counter
from itertools import count, combinations_with_replacement as mc
def aupto(lim):
  sq = filter(lambda x: x<=lim, (i**2 for i in range(int(lim**(1/2))+2)))
  s3 = filter(lambda x: 0Michael S. Branicky, Jul 01 2021

Extensions

Extended by Ray Chandler, Nov 30 2006
a(45) and beyond from Michael S. Branicky, Jul 01 2021

A115289 a(n) is the smallest number representable in exactly n ways as a sum of one triangular number and 2 squares (each of them >= 0).

Original entry on oeis.org

0, 1, 5, 10, 19, 26, 55, 46, 53, 116, 86, 128, 173, 145, 200, 170, 221, 235, 236, 305, 341, 326, 491, 425, 548, 431, 676, 530, 536, 635, 656, 851, 905, 695, 1118, 950, 1040, 1171, 1241, 1031, 1076, 1115, 1325, 1661, 1943, 1391, 1531, 1691, 1790, 1670, 2291, 2081
Offset: 1

Views

Author

Giovanni Resta, Jan 19 2006

Keywords

Examples

			a(4)=10 since 10 can be expressed in 4 ways,
10=T(4)+0^2+0^2 = T(3)+0^2+2^2 = T(1)+0^2+3^2 = T(0)+1^2+3^2 and none of the numbers from 0 to 9 can be expressed in 4 ways.

Crossrefs

Cf. A115288, A000437, A061262.

Programs

  • Mathematica
    V = Table[0, {i, 5000}]; T[n]:=n(n+1)/2; Do[a = T[i]+j^2+k^2; If[a<5000, V[[a+1]]++ ], {i, 0, 100}, {j, 0, 71}, {k, 0, j}]; Table[Position[V, z][[1, 1]]-1, {z, 60}]
Showing 1-4 of 4 results.

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