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

1,2

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

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... .

Mathematica

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

Prog

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

Crossrefs

Cf. A048760, A197680, A386470, A386471.

Similar sequences: A008833, A350390, A365683.

Sequence in context: A166140 A019555 A378997 * A382904 A243074 A304776

Adjacent sequences: A386466 A386467 A386468 * A386470 A386471 A386472

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jul 22 2025

Status

approved