A357587 If k > 1 and k divides DedekindPsi(k) then A358015(k)/2 is a term of this sequence.

1, 4, 3, 8, 12, 16, 9, 24, 32, 36, 48, 27, 64, 72, 96, 108, 128, 144, 81, 192, 216, 256, 288, 324, 384, 432, 243, 512, 576, 648, 768, 864, 972, 1024, 1152, 1296, 729, 1536, 1728, 1944, 2048, 2304, 2592, 2916, 3072, 3456, 3888, 4096, 2187, 4608, 5184, 5832, 6144

Offset

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..53.

Maple

alias(DedekindPsi = A001615):
A357587List := proc(sup) local S, A, k, j;
   S := select(n -> irem(DedekindPsi(n), n) = 0, [$2..sup]):
   A := proc(n) k := padic[ordp](n, 2); j := ifelse(irem(n, 4) = 0, k, 0);
        return 2^(j-2)*DedekindPsi(n*2^(-k)) end;
seq(A(k), k = S) end: A357587List(20000);

Mathematica

DedekindPsi[n_] := n*Times @@ (1 + 1/FactorInteger[n][[All, 1]]);
A358015[n_] := Module[{k, j}, k = IntegerExponent[n, 2]; j = If[Divisible[n, 4], k, 0]; DedekindPsi[n*2^(-k)]*2^(j - 1)];
Reap[For[k = 2, k <= 20000, k++, If[Divisible[DedekindPsi[k], k], Sow[A358015[k]/2]]]][[2, 1]] (* Jean-François Alcover, Mar 12 2024 *)

Prog

(SageMath)
from sage.arith.misc import dedekind_psi
def A357587List(sup):
    S = [n for n in srange(2, sup) if n.divides(dedekind_psi(n))]
    def A(n):
        k = valuation(n, 2)
        j = k if Integer(4).divides(n) else 0
        return 2^(j-2)*dedekind_psi(n*2^(-k))
    return [A(k) for k in S]
print(A357587List(20000))

Crossrefs

Cf. A001615, A358015, A033845, A003586.

Sequence in context: A316688 A309517 A105185 * A165739 A196132 A198179

Adjacent sequences: A357584 A357585 A357586 * A357588 A357589 A357590

Keyword

nonn

Author

Peter Luschny, Oct 26 2022

Status

approved