A300216 - OEIS

%I #32 Oct 28 2020 10:05:43

%S 1,3,14,22,28,44,46,50,56,88,92,94,112,118,166,176,184,188,198,214,

%T 224,236,294,332,334,352,358,368,376,414,428,448,454,472,500,526,664,

%U 668,694,704,716,718,726,736,752,766,846,856,882,896,908,934,944,958,1006

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

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

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

%C 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.

%H Alois P. Heinz, <a href="/A300216/b300216.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%e 1 is a term since phi(1) is 1. The prime signature of 1 is 0 or the empty set {0}.

%e 3 is a term since phi(3)=2 and both are prime, hence prime signature is {1}.

%e 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}.

%p s:= n-> sort(map(i-> i[2], ifactors(n)[2])):

%p a:= proc(n) option remember; local k; for k from 1+

%p       a(n-1) while s(k)<>s(numtheory[phi](k)) do od; k

%p     end: a(0):=0:

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Feb 28 2018

%t s[k_] := Sort[FactorInteger[k][[All, 2]]];

%t filterQ[k_] := Switch[k, 2, False, 3, True, _, s[k] == s[EulerPhi[k]]];

%t Select[Range[2000], filterQ] (* _Jean-François Alcover_, Oct 28 2020 *)

%o (PARI) isok(k) = vecsort(factor(k)[,2]) == vecsort(factor(eulerphi(k))[,2]); \\ _Michel Marcus_, Mar 09 2018

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

%K nonn

%O 1,2

%A _Torlach Rush_, Feb 28 2018