A069933 Number of k, 1<=k<=n, such that core(k) divides n, where core(x) is the squarefree part of x, the smallest integer such that x*core(x) is a square.

1, 2, 2, 3, 3, 5, 3, 4, 4, 7, 4, 8, 4, 7, 7, 6, 5, 10, 5, 10, 8, 9, 5, 11, 7, 10, 8, 11, 6, 18, 6, 9, 10, 11, 10, 15, 7, 12, 11, 14, 7, 20, 7, 13, 13, 12, 7, 16, 9, 17, 13, 15, 8, 19, 13, 16, 13, 14, 8, 27, 8, 14, 15, 13, 14, 25, 9, 16, 14, 25, 9, 21, 9, 16, 18, 17, 14, 27, 9, 20, 14, 17

Offset

1,2

Comments

Sequence does not give the number of all integers of the form core(k) dividing n since this set is infinite (for any square x^2 core(x^2)=1 divides n).

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{1<=k<=n,(n,k)=1} (-1)^bigomega(k)*floor(n/k). - Benoit Cloitre, May 07 2016

Mathematica

core[n_] := core[n] = Block[{t = Transpose@ FactorInteger@n}, Times @@ (t[[1]]^Mod[t[[2]], 2])]; a[n_] := Length@ Select[Range@n, Mod[n, core@#] == 0 &]; Array[a, 1000] (* Giovanni Resta, May 07 2016 *)

Prog

(PARI) a(n)=sum(i=1, n, if(n%core(i), 0, 1)) \\ Benoit Cloitre, May 07 2016
(PARI) a(n)=sum(k=1, n, if(gcd(n, k)-1, 0, (-1)^bigomega(k)*(n\k))) \\ Benoit Cloitre, May 07 2016
(Sage)
a = sloane.A001222
[sum([n//k*(-1)^a(k) for k in (1..n) if gcd(k, n) == 1]) for n in (1..82)] # Peter Luschny, May 08 2016

Crossrefs

Cf. A007913.

Sequence in context: A169897 A254309 A079375 * A204987 A102347 A230846

Adjacent sequences: A069930 A069931 A069932 * A069934 A069935 A069936

Keyword

easy,nonn

Author

Benoit Cloitre, May 05 2002

Status

approved