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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 (list; graph; refs; listen; history; text; internal format)
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: A019000 A305090 A071836 * A258218 A255219 A226341

Adjacent sequences:  A300213 A300214 A300215 * A300217 A300218 A300219

KEYWORD

nonn

AUTHOR

Torlach Rush, Feb 28 2018

STATUS

approved

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Last modified October 22 07:53 EDT 2021. Contains 348160 sequences. (Running on oeis4.)