A300216 Numbers k such that k and phi(k) share the same prime signature.

1, 3, 14, 22, 28, 44, 46, 50, 56, 88, 92, 94, 112, 118, 166, 176, 184, 188, 198, 214, 224, 236, 294, 332, 334, 352, 358, 368, 376, 414, 428, 448, 454, 472, 500, 526, 664, 668, 694, 704, 716, 718, 726, 736, 752, 766, 846, 856, 882, 896, 908, 934, 944, 958, 1006

Offset: 1

Author

Torlach Rush, Feb 28 2018

Keywords

• nonn

Comments

a(1) = 1 and a(2) = 3 are the only odd terms of this sequence.

For n > 2 there are no squarefree a(n) with an odd number of prime factors.

a(8) = 50 is the first even term such that 2*a(n) is not an element. The smallest multiple of a(8), a term of the sequence is a(35) = 10*a(8) = 500.

Examples

			1 is a term since phi(1) is 1. The prime signature of 1 is 0 or the empty set {0}.
3 is a term since phi(3)=2 and both are prime, hence prime signature is {1}.
14 is a term since phi(14)=6 and 14 and 6 are both the product of two distinct primes and the prime signature is {1,1}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A046523, A244733, A280927.

Programs

Maple

s:= n-> sort(map(i-> i[2], ifactors(n)[2])):
a:= proc(n) option remember; local k; for k from 1+
      a(n-1) while s(k)<>s(numtheory[phi](k)) do od; k
    end: a(0):=0:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 28 2018

Mathematica

s[k_] := Sort[FactorInteger[k][[All, 2]]];
filterQ[k_] := Switch[k, 2, False, 3, True, _, s[k] == s[EulerPhi[k]]];
Select[Range[2000], filterQ] (* Jean-François Alcover, Oct 28 2020 *)

PARI

isok(k) = vecsort(factor(k)[,2]) == vecsort(factor(eulerphi(k))[,2]); \\ Michel Marcus, Mar 09 2018

Formula

{ k | A046523(k) = A046523(A000010(k)) }.

For all n >= 1: tau(a(n)) = tau(phi(a(n))).

For all n >= 1: tau(a(n)) = tau(phi(a(n))) = 4 => sigma(a(n)) = 2*sigma(phi(a(n))).