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

Author

Amiram Eldar, Jul 22 2025

Keywords

• nonn
• easy
• mult

Comments

The largest term in A197680 that divides n.

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

Links

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

Crossrefs

Cf. A048760, A197680, A386470, A386471.

Similar sequences: A008833, A350390, A365683.

Programs

Mathematica

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

PARI

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

Formula

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... .