login
A300216
Numbers k such that k and phi(k) share the same prime signature.
5
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
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
KEYWORD
nonn
AUTHOR
Torlach Rush, Feb 28 2018
STATUS
approved