A273190 a(n) is the number of nonnegative m < n for which m + n is a perfect square.
0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4
Offset: 0
Examples
a(1) = 1 because 1 + 0 is a perfect square. a(2) = 0 because neither 2 + 0 nor 2 + 1 are perfect squares. a(5) = 1 because 5 + 4 is a perfect square. a(9) = 2 because 9 + 0 and 9 + 7 are perfect squares.
Links
- Peter Kagey, Table of n, a(n) for n = 0..10000
Programs
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Haskell
a273190 n = length $ filter (>=n) $ takeWhile (< 2 * n) $ map (^2) [1..] -- Peter Kagey, May 25 2016
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Java
int n = 100; int[] terms = new int[n]; for (int i = 0; i < n; i++) { for (int j = 0; j < i; j++) { if (Math.sqrt(i+j) == Math.floor(Math.sqrt(i+j))) { terms[i]++; } } System.out.print(terms[i] + ", "); } -
Mathematica
Table[Count[Range[0, n - 1], m_ /; IntegerQ@ Sqrt[m + n]], {n, 0, 120}] (* Michael De Vlieger, May 18 2016 *) -
PARI
a(n) = sum(k=0, n-1, issquare(n+k)); \\ Michel Marcus, May 18 2016
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Python
from gmpy2 import isqrt def A273190(n): return isqrt(2*n-1)-isqrt(n-1) if n > 0 else 0 # Chai Wah Wu, May 25 2016
Formula
a(n) = floor(sqrt(2*n-1))-floor(sqrt(n-1)) for n > 0. - Chai Wah Wu, May 25 2016
a(n) = Sum_{i=1..n} c(2*n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020