A365683 The largest exponentially squarefree divisor of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Author

Amiram Eldar, Sep 15 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A058035 at n = 32.

The number of these divisors is A365680(n) and their sum is A365682(n).

Links

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

Crossrefs

Cf. A070321, A209061, A365680, A365682.

Programs

Mathematica

f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k--]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(e) = {my(k = e); while(!issquarefree(k), k--); k;};
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Formula

Multiplicative with a(p^e) = p^A070321(e).

a(n) <= n, with equality if and only if n is exponentially squarefree number (A209061).

Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.487850776747... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = A070321(k) and f(0) = 0.