A065155 Numbers whose cototient of totient is strictly greater than totient of cototient.

5, 7, 9, 11, 13, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 106

Offset

1,1

Comments

All prime numbers greater than 3 are in this sequence. Given p prime, it is easy to see that phi(p) = p - 1 and therefore the cototient of p is 1. For p > 3, phi(p) = 2q, with q > 1 an odd number not necessarily prime. Then 2q - 1 > 2q - phi(2q) > 1. - Alonso del Arte, Jun 02 2013

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

Formula

Numbers k such that phi(k) - phi(phi(k)) > phi(k - phi(k)) or A051953(A000010(k)) > A000010(A051953(k)).

Example

11 is in the sequence, since phi(11) = 10, cototient(11) = 1, phi(1) = 1 < cototient(10) = 4.

Mathematica

eu[n_] := EulerPhi[n]; co[n_] := n - EulerPhi[n]; A065152 = Table[co[eu[w]] - eu[co[w]], {w, 1, 256}]; Flatten[Position[Sign[A065152], 1]]
(* alternative program *)
Select[Range[100], EulerPhi[#] - EulerPhi[EulerPhi[#]] > EulerPhi[# - EulerPhi[#]] &] (* Alonso del Arte, Jun 02 2013 *)

Prog

(PARI) { n=0; for (m = 2, 10^9, t=eulerphi(m); c=m - t; if (t - eulerphi(t) > eulerphi(c), write("b065155.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 13 2009

Crossrefs

Cf. A000010, A051953, A065152, A065153, A065154.

Sequence in context: A031102 A005763 A035034 * A230203 A339910 A303578

Adjacent sequences: A065152 A065153 A065154 * A065156 A065157 A065158

Keyword

nonn,changed

Author

Labos Elemer, Oct 19 2001

Status

approved