A308135 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

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).

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)

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

EXAMPLE

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.

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]

CROSSREFS

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

Cf. A013661, A065487.

Sequence in context: A320832 A034460 A063919 * A072815 A348979 A080304

Adjacent sequences: A308132 A308133 A308134 * A308136 A308137 A308138

KEYWORD

nonn

AUTHOR

Amiram Eldar and Paolo P. Lava, May 14 2019

STATUS

approved