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A367988 The sum of the divisors of the square root of the largest unitary divisor of n that is a square.

%I A367988 #10 Apr 20 2025 07:32:35
%S A367988 1,1,1,3,1,1,1,1,4,1,1,3,1,1,1,7,1,4,1,3,1,1,1,1,6,1,1,3,1,1,1,1,1,1,
%T A367988 1,12,1,1,1,1,1,1,1,3,4,1,1,7,8,6,1,3,1,1,1,1,1,1,1,3,1,1,4,15,1,1,1,
%U A367988 3,1,1,1,4,1,1,6,3,1,1,1,7,13,1,1,3,1,1
%N A367988 The sum of the divisors of the square root of the largest unitary divisor of n that is a square.
%H A367988 Amiram Eldar, <a href="/A367988/b367988.txt">Table of n, a(n) for n = 1..10000</a>
%H A367988 Vaclav Kotesovec, <a href="/A367988/a367988.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A367988 Multiplicative with a(p^e) = (p^(e/2+1)-1)/(p-1) if e is even and 1 otherwise.
%F A367988 a(n) = A000203(A071974(n)).
%F A367988 a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
%F A367988 Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1)).
%F A367988 From _Vaclav Kotesovec_, Apr 20 2025: (Start)
%F A367988 Let f(s) = Product_{p prime} (1 - 1/((p^s + 1)*p^(2*s - 1))).
%F A367988 Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
%F A367988 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
%F A367988 f(1) = A065463 = Product_{p prime} (1 - 1/(p*(1+p))) = 0.704442200999165592736603350326637210188586431417098049414226842591097056682...
%F A367988 f'(1) = f(1) * Sum_{p prime} (3*p+2)*log(p)/((p+1)*(p^2+p-1)) = f(1) * 1.167129912223800181472507785468113632129480568043855995406075158923507536957...
%F A367988 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A367988 f[p_, e_] := If[EvenQ[e], (p^(e/2 + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367988 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, (f[i,1]^(f[i,2]/2 + 1) - 1)/(f[i,1] - 1)));}
%Y A367988 Cf. A000203, A071974, A268335, A351568, A367987.
%Y A367988 Cf. A065463.
%K A367988 nonn,easy,mult
%O A367988 1,4
%A A367988 _Amiram Eldar_, Dec 07 2023