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

1,2

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.

Links

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

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))).

Example

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}.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A305090 A071836 A361909 * A258218 A255219 A226341

Adjacent sequences: A300213 A300214 A300215 * A300217 A300218 A300219

Keyword

nonn

Author

Torlach Rush, Feb 28 2018

Status

approved