1,2

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

The unitary convolution of A047994 (the unitary totient phi) with A010055 (the characteristic function of 1 and prime powers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A010055(n/d).

a(n) = uphi(n) * (1 + Sum_{p^e || n} (1/(p^e-1))), where uphi = A047994, and p^e || n denotes that the prime power p^e unitarily divides n (i.e., p^e divides n but p^(e+1) does not divide n).

a(n) = A385198(n) + A047994(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.96700643911290683406......, c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = (1 + Sum_{p prime}(1/(p^2+p-1))) = 1.37272644617447080939... .

For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 5 of the values are either 1 or a prime power, and therefore a(6) = 5.

f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct) * (1 + Total[1/f @@@ fct])]; Array[a, 100]

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1) * (1 + sum(i = 1, #f~, 1/(f[i, 1]^f[i, 2] - 1))); }

The unitary analog of A131233.

Cf. A000961, A010055, A065463, A077610, 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), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), A385196 (prime), A385197 (noncomposite), A385198 (prime power), this sequence (1 or prime power).

Sequence in context: A131233 A136623 A031218 * A357004 A357005 A267508

Adjacent sequences: A385196 A385197 A385198 * A385200 A385201 A385202

nonn,easy

Amiram Eldar, Jun 21 2025

approved