A353219 -id:A353219 - cp's OEIS frontend

A353202 Positive integers which can be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 45, 46, 49, 50, 51, 52, 53, 54, 58, 59, 61, 62, 64, 65, 66, 68, 69, 72, 73, 74, 75, 80, 81, 82, 83, 85, 86, 89, 90, 91, 97, 98, 99, 100

Offset: 1

Eric Fox, Apr 30 2022

nonn

These are the numbers which can be a clue in a Tasquare puzzle.

Tasquare also known as Tasukuea.

			A 9 clue can be satisfied in multiple ways:
  OOO  OO OO
  OOO  OO9OO
  OOO9   O

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Cross+A, Tasquare rules
Nikoli, Tasukuea rules [broken link]

Complement of A353219.

Cf. A001481.

q:= proc(n) local i; for i to isqrt(n) do if ormap(issqr,
      [n-i^2, n-i^2-1]) then return true fi od: false
    end:
select(q, [$1..100])[];  # Alois P. Heinz, Apr 30 2022

q[n_] := Module[{i}, For[i = 1, i <= Sqrt[n], i++, If[AnyTrue[ {n-i^2, n-i^2-1}, IntegerQ@Sqrt[#]&], Return[True]]]; False];
Select[Range[100], q] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)

A363769 Integers k such that the number of binary partitions of 2k is not a sum of three squares.

Original entry on oeis.org

10, 18, 34, 40, 58, 66, 72, 90, 106, 114, 130, 136, 154, 160, 170, 178, 202, 210, 226, 232, 250, 258, 264, 282, 288, 298, 306, 330, 338, 354, 360, 378, 394, 402, 418, 424, 442, 450, 456, 474, 490, 498, 514, 520, 538, 544, 554, 562, 586, 594, 610, 616, 634, 640, 650, 658, 674, 680, 698

Offset: 1

Maciej Ulas, Jun 21 2023

An infinite sequence.

			a(1)=10 because each b(20)=60 is not a sum of three squares and for i=1, ..., 9, the numbers b(2)=2, b(4)=4, b(6)=6, b(8)=10, b(10)=14, b(12)=20, b(14)=26, b(16)=36, b(18)=46 are sums of three squares, where b(i) is the number of binary partitions of n.

Bartosz Sobolewski and Maciej Ulas, Values of binary partition function represented by a sum of three squares, arXiv:2211.16622 [math.NT], 2023.

Cf. A018819, A353219, A010060 (0 -> 1 & 1 -> -1).

bin[n_] :=
 bin[n] =
  If[n == 0, 1,
   If[Mod[n, 2] == 0, bin[n - 1] + bin[n/2],
    If[Mod[n, 2] == 1, bin[n - 1]]]];
A := {}; Do[
 If[Mod[bin[2 n]/4^IntegerExponent[bin[2 n], 4], 8] == 7,
  AppendTo[A, n]], {n, 1000}];
A

Numbers of the form {2^(2*k+1)*(8*r+2*t_{r}+3): k, r nonnegative integers} and t_{r} is r-th term of the Prouhet-Thue-Morse sequence on the alphabet {-1, +1}, i.e., t_{r} = (-1)^{s_{2}(r)}, where s_{2}(r) is the sum of binary digits of r. We have t_{r} = (-1)^A010060(r).

cp's OEIS Frontend

A353202 Positive integers which can be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

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