1,2

First differs from A383763 at n = 32.

The number of these divisors is A365499(n), and the largest of them is A385007(n).

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

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.

Multiplicative with a(p^e) = p^e + 1 for e <= 3, and a(p^e) = 1 for e >= 4.

a(n) = 1 if and only if n is 4-full (A036967).

a(n) <= A034448(n), with equality if and only if n is biquadratefree.

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-3) - 1/p^(3*s-2) - 1/p^(4*s-3)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2 + p) - 1/p^4) = 1.27769267395905900191... .

f[p_, e_] := If[e < 4, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2] + 1, 1)); }

The unitary analog of A385006.

Cf. A036967, A046100, A365499, A385007.

The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), this sequence (biquadratefree).

Sequence in context: A346613 A378433 A378434 * A383763 A034448 A365211

Adjacent sequences: A385046 A385047 A385048 * A385050 A385051 A385052

nonn,easy,mult

Amiram Eldar, Jun 16 2025

approved