A069823 Nonprime numbers k for which there is no x < k such that phi(x) = phi(k).

1, 15, 25, 35, 51, 65, 69, 81, 85, 87, 121, 123, 129, 141, 143, 159, 161, 177, 185, 187, 203, 213, 235, 247, 249, 253, 255, 265, 267, 275, 289, 299, 301, 309, 321, 323, 339, 341, 343, 361, 393, 403, 415, 425, 447, 485, 489, 501, 519, 527, 529, 535, 537, 551

Offset: 1

Author

Benoit Cloitre, Apr 28 2002

Keywords

• easy
• nonn

Comments

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

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

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

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A081373, A067004, A000010.

Programs

Mathematica

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]
(* Second program: *)
Union@ Select[Values[PositionIndex@ Array[EulerPhi, 600]][[All, 1]], ! PrimeQ@ # &] (* Michael De Vlieger, Dec 31 2017 *)

PARI

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