OFFSET
1,2
COMMENTS
The number of these divisors is A365498(n).
REFERENCES
D. Suryanarayana, The number and sum of k-free intergers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, A survey on k-freeness, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.
FORMULA
Multiplicative with a(p^e) = p^e + 1 for e <= 2, and a(p^e) = 1 for e >= 3.
a(n) = 1 if and only if n is cubefull (A036966).
a(n) <= A034448(n), with equality if and only if n is cubefree.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) - 1/p^(3*s-2)).
Sum_{j=1..n} a(j) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^3 + 1/(p^2 + p)) = 1.16545286600957717104.... .
MATHEMATICA
f[p_, e_] := If[e < 3, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2] + 1, 1)); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Mar 16 2024
STATUS
approved