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.

A299796 Largest number k that is not the sum of 3 squares and a nonnegative (2n+5)-th power.

Original entry on oeis.org

2160, 19568, 177136, 1594304, 14348784, 129140160, 1162261440, 10460353136, 94143178736, 847288609392, 7625597484912, 68630377364848, 617673396283888, 5559060566555504, 50031545098999664, 450283905890997360, 4052555153018976240, 36472996377170786288
Offset: 1

Views

Author

XU Pingya, Feb 19 2018

Keywords

Comments

Subsequence of A004215.

Examples

			2160 is not the sum of 3 squares and a nonnegative 7th power, and all integers larger than 2160 can be represented as the sum of 3 squares and a nonnegative 7th power (see A297970). Thus a(1) = 2160.

Links

Crossrefs

Cf. A004215, A022552, A297970.

Programs

  • Mathematica
    a[n_]:= Max@Table[4^k (8 Floor[3^(2 n + 5) / 2^(2 k + 3) - 7 / 8] + 7),  {k, 2, n + 1}] Table[a[n], {n, 18}] (* XU Pingya, Nov 30 2018 *)

Formula

a(n) = max_{2<=i<=n+1, j>=0} {4^i*(8j+7) : 4^i*(8j+7) < 3^(2n+5)}.

A296579 Numbers that are not the sum of 3 squares and a nonnegative 9th power.

Original entry on oeis.org

112, 240, 368, 448, 496, 624, 752, 880, 960, 1008, 1136, 1264, 1392, 1472, 1520, 1648, 1776, 1904, 1984, 2032, 2160, 2288, 2416, 2496, 2544, 2672, 2800, 2928, 3008, 3056, 3184, 3312, 3440, 3520, 3568, 3696, 3824, 3952, 4032, 4080, 4208, 4336, 4464, 4544, 4592
Offset: 1

Views

Author

XU Pingya, Jan 30 2018

Keywords

Comments

a(n) consists of the number of forms 16*(8i + 7) (0 <= i <= 152) and 64*(8j + 7) (0 <= j <= 37).
The last term in this sequence is a(191) = 19568 = 16*(8*152 + 7) (see A297970).

Links

Crossrefs

Finite subsequence of A004215.
A297970 is a subsequence.
Cf. A004771, A022552, A022557, A022561, A022566, A111151.

Programs

  • Mathematica
    t1=Table[4^2*(8j+7), {j,0,152}];
t2=Table[4^3*(8j+7), {j,0,37}];
t=Union[t1, t2]

A322122 Number of positive integers that are not the sum of 3 squares and a nonnegative (2n + 1)-st power.

Original entry on oeis.org

0, 0, 17, 191, 1816, 16541, 149320, 1344880, 12106148, 108960347, 980654368, 8825914594, 79433288194, 714899721602, 6434097782077, 57906880685893, 521161927629204, 4690457351939133, 42214116174823814, 379927045590000351, 3419343410347321718
Offset: 1

Views

Author

XU Pingya, Nov 27 2018

Keywords

Links

Crossrefs

Cf. A004215, A297788, A273915, A297970, A296579, A299796.

Programs

  • Mathematica
    a[n_]:=Sum[Floor[3^(2n+1)/2^(2k+3)-7/8]+1, {k,2,n-1}];
Table[a[n], {n,21}]

Formula

a(n) = Sum_{k=2..n-1} (floor(3^(2n+1)/2^(2k+3)-7/8) + 1).

A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 3, 2, 2, 3, 4, 4, 3, 3, 3, 2, 2, 3, 5, 5, 4, 3, 3, 2, 2, 3, 5, 6, 4, 4, 3, 3, 2, 3, 5, 5, 5, 4, 5, 3, 3, 4, 5, 5, 3, 4, 4, 3, 2, 3, 6, 7, 6, 5, 6, 5, 4, 3, 4, 5, 3, 4, 4, 4, 3, 4, 7, 7, 6, 5, 5, 3, 3, 4, 7, 7, 6, 5, 4, 3, 2, 5, 7, 8, 5, 5, 6
Offset: 0

Views

Author

XU Pingya, Feb 09 2018

Keywords

Comments

4^i(8j + 7) - 1^7 - 1^7 == 5 (mod 8) (when i = 0), or 2 (when i = 1), or 6 (when i >= 2). Thus, each nonnegative integer can be written as a sum of three squares and two nonnegative 7th powers; i.e., a(n) > 0.
More generally, each nonnegative integer can be written as a sum of three squares and a nonnegative k-th power and a nonnegative m-th power.

Examples

			7 = 0^2 + 1^2 + 2^2 + 1^7 + 1^7 = 1^2 + 1^1 + 2^2 + 0^7 + 1^7, a(7) = 2.
10 = 0^2 + 0^2 + 3^2 + 0^7 + 1^7 = 0^2 + 1^1 + 3^2 + 0^7 + 0^7 = 0^2 + 2^2 + 2^2 + 1^7 + 1^7 = 1^2 + 2^1 + 2^2 + 0^7 + 1^7, a(10) = 4.

Links

Crossrefs

Cf. A000164, A002635, A004215, A004771, A297970.

Programs

  • Mathematica
    a[n_]:=Sum[If[x^2+y^2+z^2+u^7+v^7==n, 1, 0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {u,0,(n-x^2-y^2-z^2)^(1/7)}, {v,u,(n-x^2-y^2-z^2-u^7)^(1/7)}]
Table[a[n], {n,0,86}]
Showing 1-4 of 4 results.

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