A367988 -id:A367988 - cp's OEIS frontend

A367987 The number of square divisors of the largest unitary divisor of n that is a square.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 07 2023

Also, the number of divisors of the square root of the largest unitary divisor of n that is a square.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A046951, A071974, A268335, A335762, A350388, A365401, A367988.

f[p_, e_] := If[EvenQ[e], e/2 + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, 1, x/2+1), factor(n)[, 2]));

Multiplicative with a(p^e) = e/2 + 1 if e is even and 1 otherwise.
a(n) = A046951(A350388(n)).
a(n) = A000005(A071974(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/((p-1)*(p+1)^2)) = 1.450032... (A335762).

A367989 The sum of square divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1

Offset: 1

Amiram Eldar, Dec 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A035316, A268335, A350388, A367988.

f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, (f[i,1]^(f[i,2] + 2) - 1)/(f[i,1]^2 - 1)));}

a(n) = A035316(A350388(n)).
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) 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-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2)) = 0.69451968056653021193... .

cp's OEIS Frontend

A367987 The number of square divisors of the largest unitary divisor of n that is a square.

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