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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).

%I A385199 #7 Jun 22 2025 16:33:31
%S A385199 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,
%T A385199 27,27,29,22,31,32,32,33,34,35,37,37,38,39,41,32,43,43,44,45,47,47,49,
%U A385199 49,50,51,53,53,54,55,56,57,59,50,61,61,62,64,64,52,67,67
%N 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).
%H A385199 Amiram Eldar, <a href="/A385199/b385199.txt">Table of n, a(n) for n = 1..10000</a>
%F A385199 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).
%F A385199 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).
%F A385199 a(n) = A385198(n) + A047994(n).
%F A385199 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... .
%e A385199 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.
%t A385199 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]
%o A385199 (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)));}
%Y A385199 The unitary analog of A131233.
%Y A385199 Cf. A000961, A010055, A065463, A077610, A384047.
%Y A385199 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).
%K A385199 nonn,easy
%O A385199 1,2
%A A385199 _Amiram Eldar_, Jun 21 2025