A366764 -id:A366764 - cp's OEIS frontend

A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 5, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 21 2023

The number of terms of A337050 that divide n.

The sum of these divisors is A366764(n), and the largest of them is A366765(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A004709, A005117, A034444, A337050, A366764, A366765.

Cf. A330596.

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

a(n) = vecprod(apply(x -> max(x, 2), factor(n)[, 2]));

Multiplicative with a(p^e) = max(e, 2);
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034444(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3) = A330596 = 0.74853525968236356464421504863791060164164034300532440451585279392592558689...,
f'(1) = f(1) * Sum_{p prime} (2*p-3)*log(p)/(p^3-p+1) = f(1) * 0.560697508735949606541137451100554649565120075155278833396722097786365686597...
and gamma is the Euler-Mascheroni constant A001620. (End)

A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Oct 21 2023

The largest term of A337050 that divides n.

The number of these divisors is A366763(n), and their sum is A366764(n).

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

Cf. A337050, A366763, A366764.

Similar sequences: A055231, A057521, A008833, A350390.

f[p_, e_] := p^If[e < 3, 1, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1] ^ if(f[i, 2] < 3, 1, f[i, 2]));}

Multiplicative with a(p^e) = p if e <= 2 and p^e otherwise.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 2/p^3 - 1/p^4) = 0.83234421330425224469... .

cp's OEIS Frontend

A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

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