A351520 Number of numbers <= n that are either squarefree, a divisor of n, or both.

1, 2, 3, 4, 4, 5, 6, 8, 7, 7, 8, 10, 9, 10, 11, 14, 12, 14, 13, 15, 14, 15, 16, 20, 17, 17, 19, 19, 18, 19, 20, 24, 21, 22, 23, 28, 24, 25, 26, 30, 27, 28, 29, 31, 31, 30, 31, 37, 32, 33, 32, 34, 33, 37, 34, 38, 35, 36, 37, 41, 38, 39, 41, 44, 40, 41, 42, 44, 43, 44, 45, 53, 46, 47, 49

Offset: 1

Author

Wesley Ivan Hurt, Feb 12 2022

Keywords

• nonn

Examples

			a(10) = 7; There are 7 numbers less than or equal to 10 that are either squarefree, a divisor of 10, or both. The numbers are 1,2,3,5,6,7,10.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005 (tau), A013928, A034444.

Programs

Maple

f:= proc(n) nops(select(t -> n mod t = 0 or numtheory:-issqrfree(t), [$1..n])) end proc:
map(f, [$1..100]); # Robert Israel, Dec 09 2024

Mathematica

Module[{nn=80,sf},sf=Select[Range[nn],SquareFreeQ[#]&];Table[Length[Union[Select[sf,#<= n&],Divisors[n]]],{n,nn}]] (* Harvey P. Dale, Jul 03 2023 *)

Formula

a(n) = tau(n) + Sum_{k=1..n} mu(k)^2 - Sum_{d|n} mu(d)^2.

a(n) = A000005(n) + A013928(n+1) - A034444(n).