VOOZH about

URL: https://oeis.org/A385199

⇱ A385199 - OEIS


login
A385199
The number of integers k from 1 to n such that the greatest divisor of k that is either 1 or a prime power (A000961).
5
1, 2, 3, 4, 5, 5, 7, 8, 9, 9, 11, 11, 13, 13, 14, 16, 17, 17, 19, 19, 20, 21, 23, 23, 25, 25, 27, 27, 29, 22, 31, 32, 32, 33, 34, 35, 37, 37, 38, 39, 41, 32, 43, 43, 44, 45, 47, 47, 49, 49, 50, 51, 53, 53, 54, 55, 56, 57, 59, 50, 61, 61, 62, 64, 64, 52, 67, 67
OFFSET
1,2
LINKS
FORMULA
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... .
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 either 1 or a prime power, 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[1/f @@@ fct])]; 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~, 1/(f[i, 1]^f[i, 2] - 1))); }
CROSSREFS
The unitary analog of A131233.
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
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jun 21 2025
STATUS
approved