1,2

First differs A372671 from at n = 25.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Multiplicative with a(p^e) = p^e if p <= 3, and p^e-1 if p >= 5.

a(n) = n * A047994(n) / A384058(n).

a(n) = A047994(A065330(n)) * A065331(n).

Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1-1/2^s)/(1-1/2^(s-1)+1/2^(2*s-1))) * ((1-1/3^s)/(1-2/3^s+1/3^(2*s-1))) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)).

Sum_{k=1..n} a(k) ~ (36/55) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.

In general, the average order of the number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a p-smooth number (i.e., not divisible by any prime larger than the prime p) is (1/2) * Product_{q prime <= p} (1 + 1/(q^2+q-1)) * A065463 * n^2.

f[p_, e_] := p^e - If[p < 5, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - if(f[i, 1] < 5, 0, 1)); }

Unitary analog of A372671.

Cf. A003586, A065330, A065331, A065463, A372671, A384046, A384047.

The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), this sequence (3-smooth), A384058 (5-rough).

Sequence in context: A102441 A102440 A372671 * A124815 A081328 A179276

Adjacent sequences: A384054 A384055 A384056 * A384058 A384059 A384060

nonn,easy,mult

Amiram Eldar, May 18 2025

approved