A308135 Sum of non-coreful divisors of n.

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 10, 1, 10, 9, 1, 1, 15, 1, 12, 11, 14, 1, 18, 1, 16, 1, 14, 1, 42, 1, 1, 15, 20, 13, 19, 1, 22, 17, 20, 1, 54, 1, 18, 18, 26, 1, 34, 1, 33, 21, 20, 1, 42, 17, 22, 23, 32, 1, 78, 1, 34, 20, 1, 19, 78, 1, 24, 27, 74, 1, 27, 1, 40

Offset: 1

Author

Amiram Eldar and Paolo P. Lava, May 14 2019

Keywords

• nonn

Comments

Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Examples

			a(15) = 9. Prime factors of 15 are 3, 5 and its divisors are 1, 3, 5, 15. The non-coreful divisors are 1, 3, 5 and their sum is 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Crossrefs

Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308127.

Cf. A013661, A065487.

Programs

Maple

with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
sigma(k)-a*sigma(k/a); end: seq(P(i),i=1..74);

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 0; a[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Array[a, 100]

Formula

a(n) = A000203(n) - A057723(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065487 = 0.413642... . - Amiram Eldar, Dec 08 2023