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

1,2

Links

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

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.

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]

Prog

(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])); }

Crossrefs

Cf. A067535, A209061, A365685.

Sequence in context: A023769 A023796 A032950 * A309127 A248502 A250039

Adjacent sequences: A365681 A365682 A365683 * A365685 A365686 A365687

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 15 2023

Status

approved