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Nonprime numbers k for which there is no x < k such that phi(x) = phi(k).
2

%I #32 May 27 2024 07:18:41

%S 1,15,25,35,51,65,69,81,85,87,121,123,129,141,143,159,161,177,185,187,

%T 203,213,235,247,249,253,255,265,267,275,289,299,301,309,321,323,339,

%U 341,343,361,393,403,415,425,447,485,489,501,519,527,529,535,537,551

%N Nonprime numbers k for which there is no x < k such that phi(x) = phi(k).

%C If p is prime there is no x < p such that phi(x) = phi(p) = p-1 since phi(x) < p-1.

%C Nonprime numbers k such that A081373(k)=1; i.e., the number of numbers not exceeding k, and with identical phi value to that of k, equals one. - _Labos Elemer_, Mar 24 2003

%C For 1 < n, if a(n) is squarefree, then phi(a(n)) is nonsquarefree. The converse is also true: for 1 < n, if phi(a(n)) is squarefree then a(n) is nonsquarefree. - _Torlach Rush_, Dec 26 2017

%H Amiram Eldar, <a href="/A069823/b069823.txt">Table of n, a(n) for n = 1..10000</a>

%e k=25, a nonprime; phi values for k <= 25 are {1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20}; no phi(k) except phi(25) equals 20, A081373(25)=1, so 25 is a term.

%t f[x_] := EulerPhi[x] fc[x_] := Count[Table[f[j]-f[x], {j, 1, x}], 0] t1=Flatten[Position[Table[fc[w], {w, 1, 1000}], 1]] t2=Flatten[Position[PrimeQ[t1], False]] Part[t1, t2]

%t (* Second program: *)

%t Union@ Select[Values[PositionIndex@ Array[EulerPhi, 600]][[All, 1]], ! PrimeQ@ # &] (* _Michael De Vlieger_, Dec 31 2017 *)

%o (PARI) for(s=1,600,if((1-isprime(s))*abs(prod(i=1,s-1,eulerphi(i)-eulerphi(s)))>0, print1(s,",")))

%Y Cf. A081373, A067004, A000010.

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Apr 28 2002