

A065155


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


3



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
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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 n such that phi(n)  phi(phi(n)) > phi(n  phi(n)) orA051953(A000010(n)) > A000010(A051953(n)).


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]] (* Labos *)
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 A303578 A282057
Adjacent sequences: A065152 A065153 A065154 * A065156 A065157 A065158


KEYWORD

nonn


AUTHOR

Labos Elemer, Oct 19 2001


STATUS

approved



