A386470 -id:A386470 - cp's OEIS frontend

A386469 The largest divisor of n whose exponents in its prime factorization are squares.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 16, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 16, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Jul 22 2025

The largest term in A197680 that divides n.

The number of these divisors is A386470(n) and their sum is A386471(n).

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

Cf. A048760, A197680, A386470, A386471.

Similar sequences: A008833, A350390, A365683.

f[p_, e_] := p^(Floor[Sqrt[e]]^2); 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]^(sqrtint(f[i, 2])^2)); }

Multiplicative with a(p^e) = p^A048760(e).
a(n) <= n, with equality if and only if n is in A197680.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} Sum_{k>=2} (1/p^(k^2-1) - 1/p^(k^2-2)) = 0.74491327356409794092... .

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... .

cp's OEIS Frontend

A386469 The largest divisor of n whose exponents in its prime factorization are squares.

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