A390692 - OEIS

%I #9 Nov 15 2025 08:16:17

%S 1,2,1,1,5,2,1,2,1,10,11,1,1,2,5,1,17,2,1,5,1,22,23,2,1,2,1,1,29,10,1,

%T 2,11,34,5,1,1,2,1,10,41,2,1,11,5,46,47,1,1,2,17,1,53,2,55,2,1,58,59,

%U 5,1,2,1,1,5,22,1,17,23,10,71,2,1,2,1,1,11,2,1,5,1,82,83,1,85,2,29,22,89,10,1,23,1,94,5,2,1,2,11,1

%N a(n) is the smallest positive integer such that a(n) * n is a Loeschian number (A003136).

%H Amiram Eldar, <a href="/A390692/b390692.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^(e mod 2) if p == 2 (mod 3), and a(p^e) = 1 otherwise.

%F a(n) = n / A390691(n).

%F a(n) = n if and only if n is squarefree and all the prime factors of n are == 2 (mod 3).

%F a(n) = 1 if and only if n is in A003136.

%t f[p_, e_] := If[Mod[p, 3] == 2, p^Mod[e, 2], 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1]%3 == 2, f[i, 1]^(f[i, 2] % 2), 1)); }

%Y Cf. A003136, A363340, A390691.

%K nonn,mult,easy

%O 1,2

%A _Amiram Eldar_, Nov 15 2025