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A367990 Sum of the squarefree divisors of the largest unitary divisor of n that is a square.

%I A367990 #10 Apr 20 2025 05:40:53
%S A367990 1,1,1,3,1,1,1,1,4,1,1,3,1,1,1,3,1,4,1,3,1,1,1,1,6,1,1,3,1,1,1,1,1,1,
%T A367990 1,12,1,1,1,1,1,1,1,3,4,1,1,3,8,6,1,3,1,1,1,1,1,1,1,3,1,1,4,3,1,1,1,3,
%U A367990 1,1,1,4,1,1,6,3,1,1,1,3,4,1,1,3,1,1,1
%N A367990 Sum of the squarefree divisors of the largest unitary divisor of n that is a square.
%H A367990 Amiram Eldar, <a href="/A367990/b367990.txt">Table of n, a(n) for n = 1..10000</a>
%H A367990 Vaclav Kotesovec, <a href="/A367990/a367990.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A367990 Multiplicative with a(p^e) = p + 1 if e is even and 1 otherwise.
%F A367990 a(n) = A048250(A350388(n)).
%F A367990 a(n) = A000203(A336643(n)).
%F A367990 a(n) = A048250(n)/A367991(n).
%F A367990 a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
%F A367990 Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1)).
%F A367990 From _Vaclav Kotesovec_, Apr 20 2025: (Start)
%F A367990 Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} ((p^(2*s) - p) * (p^(2*s) + p^s + p) / ((p^s+1) * p^(3*s))).
%F A367990 Let f(s) = Product_{p prime} ((p^(2*s)-p) * (p^(2*s)+p^s+p) / ((p^s+1) * p^(3*s))).
%F A367990 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
%F A367990 f(1) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321167...
%F A367990 f'(1) = f(1) * Sum_{p prime} (7*p + 5) * log(p) / ((p-1)*(p+1)*(p+2)) = f(1) * 3.0570993566532132522378281945383016697995408795919384628849894110222383828...
%F A367990 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A367990 f[p_, e_] := If[EvenQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367990 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(!(f[i,2]%2), f[i,1]+1, 1));}
%Y A367990 Cf. A000203, A048250, A268335, A336643, A350388, A367991.
%Y A367990 Cf. A307868.
%K A367990 nonn,easy,mult
%O A367990 1,4
%A A367990 _Amiram Eldar_, Dec 07 2023