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A384656 a(n) = Sum_{k=1..n} A051903(ugcd(n,k)), where ugcd(n,k) is the greatest divisor of k that is a unitary divisor of n.

%I A384656 #8 Jun 07 2025 08:18:42
%S A384656 0,1,1,2,1,4,1,3,2,6,1,9,1,8,7,4,1,12,1,13,9,12,1,16,2,14,3,17,1,22,1,
%T A384656 5,13,18,11,24,1,20,15,22,1,30,1,25,18,24,1,27,2,28,19,29,1,32,15,28,
%U A384656 21,30,1,51,1,32,22,6,17,46,1,37,25,46,1,41,1,38,30
%N A384656 a(n) = Sum_{k=1..n} A051903(ugcd(n,k)), where ugcd(n,k) is the greatest divisor of k that is a unitary divisor of n.
%C A384656 The terms of this sequence can be calculated efficiently using the 1st formula. The value of the function f(n, k) is equal to the number of integers i from 1 to n such that the greatest divisor of k that is a unitary divisor of n is is 1 if k = 1, or k-free if k >= 2 (k-free numbers are numbers that are not divisible by a k-th power other than 1). E.g., f(n, 1) = A047994(n), f(n, 2) = A384048(n), and f(n, 3) = A384049(n).
%C A384656 The record values of a(n)/n are 1, 2, 6, 12, 60, 420, ..., i.e, 1, 2, 6, followed by twice the primorials (A088860, A097250) starting from 2*primorial(2) = 2*A002110(2) =  12. The record values of a(n)/n converge to 5/4.
%H A384656 Amiram Eldar, <a href="/A384656/b384656.txt">Table of n, a(n) for n = 1..10000</a>
%F A384656 a(n) = Sum_{k=1..A051903(n)} (n - f(n, k)) = A051903(n) * n  - Sum_{k=1..A051903(n)} f(n, k), where f(n, k) is multiplicative for a given k, with f(p^e, k) = p^e - 1 if e >= k and f(p^e, k) = p^e if e < k.
%F A384656 a(n) = 1 if and only if n is prime.
%F A384656 a(n) >= 2 if and only if n is composite.
%F A384656 a(n) >= n - A047994(n) with equality if and only if n is squarefree (A005117).
%F A384656 a(n) >= 2*n - A047994(n) - A384048(n) with equality if and only if n is cubefree that is not squarefree (i.e., n in A067259, or equivalently, A051903(n) = 2).
%F A384656 a(n) <= A384655(n) with equality if and only if n is squarefree (A005117).
%F A384656 a(n) < 5*n/4 and lim sun_{n->oo} a(n)/n = 5/4.
%F A384656 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1 - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1)))) = 0.36292303251495264373... .
%e A384656 a(4) = A051903(ugcd(4,1)) + A051903(ugcd(4,2)) + A051903(ugcd(4,3)) + A051903(ugcd(4,4)) = A051903(1) + A051903(1) + A051903(1) + A051903(4) = 0 + 0 + 0 + 2 = 2.
%t A384656 f[p_, e_, k_] := p^e - If[e < k, 0, 1]; a[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s]; a[1] = 0; Array[a, 100]
%o A384656 (PARI) a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, 1))); s);
%Y A384656 Unitary analog of A384655.
%Y A384656 Cf. A000961, A002110, A005117, A047994, A051953, A088860, A097250, A384046, A384048, A384049, A383159.
%K A384656 nonn
%O A384656 1,4
%A A384656 _Amiram Eldar_, Jun 06 2025