number.wiki

Term

Squarefree

Numbers not divisible by the square of any prime — every prime appears in the factorization at most once.

10,554 numbers tagged.

A squarefree integer is one not divisible by the square of any prime. Equivalently, every prime factor appears with exponent exactly 1. Examples: 1, 2, 3, 5, 6 (= 2·3), 7, 10 (= 2·5), 11, 13, 14, 15, 17, 19, 21, 22.

Non-examples: 4 (= 2²), 8 (= 2³), 9 (= 3²), 12 (= 2² · 3), 18 (= 2 · 3²), 25 (= 5²).

Squarefree numbers have density \(6/\pi^2 \approx 0.6079\) in the natural numbers — roughly 61% of integers are squarefree. The Möbius function \(\mu(n)\) is zero precisely on non-squarefree numbers, and \(\pm 1\) otherwise.

36,209 36,217 36,229 36,241 36,251 36,263 36,269 36,277 36,293 36,299 36,307 36,313 36,319 36,341 36,343 36,353 36,373 36,383 36,389 36,433 36,451 36,457 36,467 36,469 36,473 36,479 36,493 36,497 36,523 36,527 36,529 36,541 36,551 36,559 36,563 36,571 36,583 36,587 36,599 36,607 36,629 36,637 36,643 36,653 36,671 36,677 36,683 36,691 36,697 36,709 36,713 36,721 36,739 36,749 36,761 36,767 36,779 36,781 36,787 36,791 36,793 36,809 36,821 36,833 36,847 36,857 36,871 36,877 36,887 36,899 36,901 36,913 36,919 36,923 36,929 36,931 36,943 36,947 36,973 36,979 36,997 37,003 37,013 37,019 37,021 37,039 37,049 37,057 37,061 37,087 37,097 37,117 37,123 37,139 37,159 37,171 37,181 37,189 37,199 37,201 37,217 37,223 37,243 37,253 37,273 37,277 37,307 37,309 37,313 37,321 37,337 37,339 37,357 37,361 37,363 37,369 37,379 37,397 37,409 37,423 37,441 37,447 37,463 37,483 37,489 37,493 37,501 37,507 37,511 37,517 37,529 37,537 37,547 37,549 37,561 37,567 37,571 37,573 37,579 37,589 37,591 37,607 37,619 37,633 37,643 37,649 37,657 37,663 37,691 37,693 37,699 37,717 37,747 37,781 37,783 37,799 37,811 37,813 37,831 37,847 37,853 37,861 37,871 37,879 37,889 37,897 37,907 37,951 37,957 37,963 37,967 37,987 37,991 37,993 37,997 38,011 38,039 38,047 38,053 38,069 38,083 38,113 38,119 38,149 38,153 38,167 38,177 38,183 38,189 38,197 38,201 38,219 38,231 38,237 38,239 38,261 38,273 38,281 38,287 38,299