A385197 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 noncomposite number.

1, 2, 3, 3, 5, 5, 7, 7, 8, 9, 11, 9, 13, 13, 14, 15, 17, 16, 19, 15, 20, 21, 23, 21, 24, 25, 26, 21, 29, 22, 31, 31, 32, 33, 34, 24, 37, 37, 38, 35, 41, 32, 43, 33, 40, 45, 47, 45, 48, 48, 50, 39, 53, 52, 54, 49, 56, 57, 59, 42, 61, 61, 56, 63, 64, 52, 67, 51

Offset

1,2

Links

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

Formula

The unitary convolution of A047994 (the unitary totient phi) with A080339 (the characteristic function of noncomposite numbers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A080339(n/d).

a(n) = uphi(n) * (1 + Sum_{p || n} (1/(p-1))), where uphi = A047994, and p || n denotes that p unitarily divides n (i.e., the p-adic valuation of n is 1).

a(n) = A385196(n) + A047994(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.92334965064835578762..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = 1 + Sum_{p prime}((p^2-1)/(p^2*(p^2+p-1))) = 1.31075288978811405615... .

Example

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 noncomposite numbers, and therefore a(6) = 5.

Mathematica

f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(1 + Total[Boole[# == 1] & /@ fct[[;; , 2]]/(fct[[;; , 1]] - 1)])]; Array[a, 100]

Prog

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

Crossrefs

The unitary analog of A349338.

Cf. A047994, A065463, A077610, A080339, 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), this sequence (noncomposite), A385198 (prime power), A385199 (1 or prime power).

Sequence in context: A281354 A283313 A091937 * A239906 A239907 A113637

Adjacent sequences: A385194 A385195 A385196 * A385198 A385199 A385200

Keyword

nonn,easy

Author

Amiram Eldar, Jun 21 2025

Status

approved