A386469 -id:A386469 - cp's OEIS frontend

A386470 The number of divisors of n whose exponents in their prime factorization are squares.

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

Offset: 1

Amiram Eldar, Jul 22 2025

First differs from A365171 and A369310 at n = 32.
First differs from A365488 at n = 128.
The number of terms in A197680 that divide n.
The sum of these divisors is A386471(n) and the largest of them is A386469(n).

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

Cf. A000005, A005117, A197680, A365171, A365488, A369310, A386469, A386471.

f[p_, e_] := Floor[Sqrt[e]] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = A048760(e) + 1.

a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).

A386471 The sum of the divisors of n whose exponents in their prime factorization are squares.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 19, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 19, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 76, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 19, 84, 144, 68, 54

Offset: 1

Amiram Eldar, Jul 23 2025

The sum of the terms in A197680 that divide n.

The number of these divisors is A386470(n) and the largest of them is A386469(n).

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

Cf. A000203, A005117, A197680, A386469, A386470.

f[p_, e_] := Sum[p^(k^2), {k, 0, Floor[Sqrt[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sum(k = 0, sqrtint(f[i,2]), f[i,1]^(k^2)));}

Multiplicative with a(p^e) = Sum_{k=0..floor(sqrt(e))} p^(k^2).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + Sum_{k>=2} Sum_{i=(2*k)^2..(2*k+1)^2-1} (-1/p)^i) = 1.05459216969289486594... .

A386472 The maximum exponent in the prime factorization of the largest divisor of n whose exponents in its prime factorization are squares.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 23 2025

All the terms are by definition squares.

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

Cf. A048760, A051903, A386469.

a[n_] := Floor[Sqrt[Max[FactorInteger[n][[;; , 2]]]]]^2; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, sqrtint(vecmax(factor(n)[,2]))^2);

a(n) = A051903(A386469(n)).
a(n) = A048760(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2*k+1)*(1-1/zeta((k+1)^2)) = 1.2383138701540899647042996... .

cp's OEIS Frontend

A386470 The number of divisors of n whose exponents in their prime factorization are squares.

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A386471 The sum of the divisors of n whose exponents in their prime factorization are squares.

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Author

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Mathematica

PARI

Formula

A386472 The maximum exponent in the prime factorization of the largest divisor of n whose exponents in its prime factorization are squares.

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula