A367988 The sum of the divisors of the square root of the largest unitary divisor of n that is a square.
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, 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, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 7, 13, 1, 1, 3, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Programs
-
Mathematica
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]
-
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)));}
Formula
Multiplicative with a(p^e) = (p^(e/2+1)-1)/(p-1) if e is even and 1 otherwise.
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/((p^s + 1)*p^(2*s - 1))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A065463 = Product_{p prime} (1 - 1/(p*(1+p))) = 0.704442200999165592736603350326637210188586431417098049414226842591097056682...
f'(1) = f(1) * Sum_{p prime} (3*p+2)*log(p)/((p+1)*(p^2+p-1)) = f(1) * 1.167129912223800181472507785468113632129480568043855995406075158923507536957...
and gamma is the Euler-Mascheroni constant A001620. (End)