cp's OEIS Frontend

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).

%I A385198 #7 Jun 22 2025 16:33:37
%S A385198 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,
%T A385198 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,
%U A385198 29,1,26,1,31,14,1,16,32,1,19,24,34,1,15,1,37,26,21
%N 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).
%H A385198 Amiram Eldar, <a href="/A385198/b385198.txt">Table of n, a(n) for n = 1..10000</a>
%F A385198 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).
%F A385198 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).
%F A385198 a(n) = A385199(n) - A047994(n).
%F A385198 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... .
%e A385198 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.
%t A385198 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]
%o A385198 (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));}
%Y A385198 The unitary analog of A116512.
%Y A385198 Cf. A047994, A065463, A069513, A077610, A246655, A384047.
%Y 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: 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).
%K A385198 nonn,easy
%O A385198 1,6
%A A385198 _Amiram Eldar_, Jun 21 2025