A385198 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 prime power (A246655).
0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 5, 1, 7, 6, 1, 1, 9, 1, 7, 8, 11, 1, 9, 1, 13, 1, 9, 1, 14, 1, 1, 12, 17, 10, 11, 1, 19, 14, 11, 1, 20, 1, 13, 12, 23, 1, 17, 1, 25, 18, 15, 1, 27, 14, 13, 20, 29, 1, 26, 1, 31, 14, 1, 16, 32, 1, 19, 24, 34, 1, 15, 1, 37, 26, 21
Offset: 1
Examples
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. 3 of the values are prime powers, and therefore a(6) = 3.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
The unitary analog of A116512.
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), this sequence (prime power), A385199 (1 or prime power).
Programs
-
Mathematica
f[p_, e_] := p^e - 1; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(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) * sum(i = 1, #f~, 1/(f[i,1]^f[i,2] - 1));}
Formula
The unitary convolution of A047994 (the unitary totient phi) with A069513 (the characteristic function of prime powers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A069513(n/d).
a(n) = uphi(n) * 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).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.26256423811374124133..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = Sum_{p prime}(1/(p^2+p-1)) = 0.37272644617447080939... .