A365684 a(n) is the smallest multiple of n that is an exponentially squarefree number (A209061).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 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, 96, 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

Links

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

Crossrefs

Cf. A067535, A209061, A365685.

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^A067535(e).

a(n) = n*A365685(n).

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

Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.532814206... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = A067535(k) and f(0) = 0.