A370574 - OEIS

A370574

a(n) is the least k > 0 such that n^2 XOR k^2 is a positive square number (where XOR denotes the bitwise XOR operator), or -1 if no such k exists.

-1, -1, 4, 3, 3, 8, 24, 6, 23, 6, 44, 16, 43, 48, 36, 12, 111, 46, 180, 12, 72, 88, 9, 7, 7, 86, 20, 59, 20, 72, 56, 24, 479, 222, 20, 15, 20, 360, 15, 24, 183, 144, 13, 11, 11, 18, 943, 14, 1103, 14, 747, 172, 405, 40, 159, 31, 492, 40, 28, 144, 28, 112, 31, 48, 660

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

This sequence has similarities with A055527; here we look for triples (n, k, m) of positive integers such that n^2 XOR k^2 = m^2, there for triples such that n^2 + k^2 = m^2.

Conjecture: a(n) > 0 for any n > 2.

LINKS

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Interactive scatterplot of (x, y, z) such that x^2 XOR y^2 = z^2 and x, y, z < 2^13 [provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes]

Rémy Sigrist, Logarithmic scatterplot of the first 140000 terms

EXAMPLE

For n = 6: we have:

k 6^2 XOR k^2 Positive square?

-- ----------- ----------------

1 37 No

2 32 No

3 45 No

4 52 No

5 61 No

6 0 No

7 21 No

8 100 Yes

So a(6) = 8.

MAPLE

f:= proc(n) local k;

for k from 1 by 1 do if k <> n and issqr(MmaTranslator[Mma]:-BitXor(n^2, k^2)) then return k fi od

end proc: f(1):= -1: f(2):= -1:

map(f, [$1..100]); # Robert Israel, Mar 01 2024

MATHEMATICA

A370574[n_] := If[n <= 2, -1, Block[{k = 0}, While[++k == n || !IntegerQ[ Sqrt[BitXor[n^2, k^2]]]]; k]]; Array[A370574, 100] (* Paolo Xausa, Mar 01 2024 *)

PROG

(PARI) a(n) = { if (n <= 2, return (-1), for (k = 1, oo, if (k!=n && issquare( bitxor(n^2, k^2)), return (k)); ); ); }

(Python)

from sympy import integer_nthroot

def A370574(n):

if n <= 2: return -1

k, n_2 = 1, n**2

while True:

if (integer_nthroot(n_2 ^ k**2, 2))[1] and k != n: return k

k += 1 # Karl-Heinz Hofmann, Mar 03 2024

CROSSREFS

Cf. A055527.