login
A306927
a(n) = A001615(n) - n.
9
0, 1, 1, 2, 1, 6, 1, 4, 3, 8, 1, 12, 1, 10, 9, 8, 1, 18, 1, 16, 11, 14, 1, 24, 5, 16, 9, 20, 1, 42, 1, 16, 15, 20, 13, 36, 1, 22, 17, 32, 1, 54, 1, 28, 27, 26, 1, 48, 7, 40, 21, 32, 1, 54, 17, 40, 23, 32, 1, 84, 1, 34, 33, 32, 19, 78, 1, 40, 27, 74, 1, 72
OFFSET
1,4
COMMENTS
Analogous to A051953.
a(n) = A051953(n) if n is an element of A000961.
a(n) > A051953(n) if n is an element of A024619.
The sum of the proper divisors d of n such that n/d is squarefree. - Amiram Eldar, Sep 06 2019
FORMULA
a(n) = A001615(n) - n.
a(n) = Sum_{d|n, d<n} (mu(n/d)^2 * d). - Amiram Eldar, Sep 06 2019
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - Amiram Eldar, Dec 08 2023
EXAMPLE
0 is a term because A001615(1) - 1 = 0.
1 is a term because A001615(2) - 2 = 1.
3 is a term because A001615(9) - 9 = 3.
MATHEMATICA
a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* Amiram Eldar, Sep 06 2019 *)
PROG
(PARI) a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ Michel Marcus, Mar 18 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Torlach Rush, Mar 16 2019
STATUS
approved