A342009 Numbers k such that the arithmetic derivative of k is a multiple of phi(k).

1, 2, 4, 8, 9, 12, 15, 16, 20, 32, 36, 48, 64, 81, 108, 112, 128, 144, 180, 189, 192, 196, 225, 256, 320, 324, 400, 432, 500, 512, 528, 576, 625, 729, 768, 972, 1024, 1225, 1296, 1300, 1360, 1452, 1728, 2048, 2160, 2304, 2700, 2816, 2916, 3024, 3072, 3375, 3564, 3840, 3888, 3993, 4096, 4800, 5120, 5184, 5292, 5616, 6000

Offset: 1

Author

Antti Karttunen, Mar 11 2021

Keywords

• nonn

Comments

Numbers k for which A000010(k) is a divisor of A003415(k), or equally, k for which A173557(k) is a divisor of A342001(k).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001

Crossrefs

Cf. A000010, A003415, A166374 (subsequence), A173557, A342001, A342008.

Positions of ones in A342415.

Programs

Mathematica

Select[Range[6000], Mod[If[Abs[#] < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]], EulerPhi[#]] == 0 &] (* Michael De Vlieger, Mar 11 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA342009(n) = !(A003415(n)%eulerphi(n));