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A308135 Sum of non-coreful divisors of n. 7
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
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
Sequence in context: A320832 A034460 A063919 * A072815 A348979 A080304
KEYWORD
nonn
AUTHOR
Amiram Eldar and Paolo P. Lava, May 14 2019
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)