A365684 - OEIS

%I #11 Sep 16 2023 02:21:58

%S 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,

%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,96,49,

%U 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68

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

%H Amiram Eldar, <a href="/A365684/b365684.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^A067535(e).

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

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

%F 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.

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

%o (PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k;};

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

%Y Cf. A067535, A209061, A365685.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Sep 15 2023