%I #12 Mar 12 2024 14:51:24

%S 1,4,3,8,12,16,9,24,32,36,48,27,64,72,96,108,128,144,81,192,216,256,

%T 288,324,384,432,243,512,576,648,768,864,972,1024,1152,1296,729,1536,

%U 1728,1944,2048,2304,2592,2916,3072,3456,3888,4096,2187,4608,5184,5832,6144

%N If k > 1 and k divides DedekindPsi(k) then A358015(k)/2 is a term of this sequence.

%C Equivalently, if k is in A033845 then A358015(k)/2 is in this sequence.

%C A term of this sequence is a 3-smooth number (A003586).

%p alias(DedekindPsi = A001615):

%p A357587List := proc(sup) local S, A, k, j;

%p S := select(n -> irem(DedekindPsi(n), n) = 0, [$2..sup]):

%p A := proc(n) k := padic[ordp](n, 2); j := ifelse(irem(n, 4) = 0, k, 0);

%p return 2^(j-2)*DedekindPsi(n*2^(-k)) end;

%p seq(A(k), k = S) end: A357587List(20000);

%t DedekindPsi[n_] := n*Times @@ (1 + 1/FactorInteger[n][[All, 1]]);

%t A358015[n_] := Module[{k, j}, k = IntegerExponent[n, 2]; j = If[Divisible[n, 4], k, 0]; DedekindPsi[n*2^(-k)]*2^(j - 1)];

%t Reap[For[k = 2, k <= 20000, k++, If[Divisible[DedekindPsi[k], k], Sow[A358015[k]/2]]]][[2, 1]] (* _Jean-François Alcover_, Mar 12 2024 *)

%o (SageMath)

%o from sage.arith.misc import dedekind_psi

%o def A357587List(sup):

%o S = [n for n in srange(2, sup) if n.divides(dedekind_psi(n))]

%o def A(n):

%o k = valuation(n, 2)

%o j = k if Integer(4).divides(n) else 0

%o return 2^(j-2)*dedekind_psi(n*2^(-k))

%o return [A(k) for k in S]

%o print(A357587List(20000))

%Y Cf. A001615, A358015, A033845, A003586.

%K nonn

%O 1,2

%A _Peter Luschny_, Oct 26 2022