Squarefree Numbers

Introduction

“A square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1.”1 In other words, its prime decomposition contains no repeated factors; there is exactly one factor for each prime that appears in it. “The number 1 is squarefree by convention.”2

Number Squarefree Prime Factors
\(0\) No \(0\)
\(1\) Yes \(1\)
\(2\) Yes \(2\)
\(3\) Yes \(3\)
\(4\) No \(2^2\)
\(5\) Yes \(5\)
\(6\) Yes \(2, 3\)
\(7\) Yes \(7\)
\(8\) No \(2^3\)
\(9\) No \(3^2\)
\(10\) Yes \(2, 5\)
Why is zero (0) not a squarefree number?

Squarefree Numbers from 1 to 100

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

for (z = 1, 100, if (issquarefree(z) == 1, print1(z, “, “)))
for (n = 1, 100, if (core(n) == n, print1(n, “, “)))

See also: OEIS A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

PARI/GP issquarefree(x) vs. core(x)

for (x = 0, 20, if (issquarefree(x) == 1, print1(x, “, “))); print(“”);
for (x = 0, 20, if (core(x) == x, print1(x, “, “))); print(“”);
\(\space\)
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19,
0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19,

 

Related Sequences

Squareful Numbers

“Squareful numbers are numbers for which at least one prime factor exponent is 2, thus are not squarefree numbers, not to be confused with squarefull numbers, numbers for which each prime factor exponent is at least 2.”3

Squareful Numbers from 1 to 100

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100

for (z = 1, 100, if (issquarefree(z) == 0, print1(z^2, “, “)))

See also: OEIS A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

Squarefree Numbers Squared

Squarefree Numbers Squared from 1 to 100

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476, 5929, 6084, 6241, 6724, 6889, 7225, 7396, 7569, 7921, 8281, 8649, 8836, 9025, 9409

for (z = 1, 100, if (issquarefree(z) == 1, print1(z^2, “, “)))

References

“Squarefree numbers”. The On-Line Encyclopedia of Integer Sequences (OEIS).

Weisstein, Eric W. “Squarefree”. Wolfram MathWorld.

Wikipedia contributors. “Square-free integer”. Wikipedia.org.

 

Footnotes

  1. Wikipedia contributors. “Square-free integer”. Wikipedia.org. Accessed July 18, 2019.
  2. Weisstein, Eric W. “Squarefree”. Wolfram MathWorld. Accessed July 18, 2019.
  3. “Squareful numbers”. OEIS.org. Accessed July 18, 2019.

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