A036798 Odd numbers m such that there exists an even number k < m with phi(k) = phi(m).

105, 165, 195, 315, 495, 525, 585, 735, 825, 945, 975, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2625, 2805, 2835, 2925, 3003, 3045, 3135, 3255, 3315, 3465, 3675, 3705, 3795, 3885, 3927, 4095, 4125, 4305, 4389, 4455

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

These numbers m appear to satisfy cototient(m) > totient(m) or 2*phi(m) < m; they seem to be the missing terms mentioned in A067800. - Labos Elemer, May 08 2003

All elements in this sequence must have 2*phi(m) < m, but not the reverse. See A118700. - Franklin T. Adams-Watters, May 21 2006

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 11, 108, 1139, 11036, 111796, ... . Apparently, the asymptotic density of this sequence exists and equals 0.011... . - Amiram Eldar, Nov 21 2024

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000010, A067800, A083254.

Cf. A091495 (Odd, squarefree n such that n/phi(n) > 2).

Cf. A118700, A119434.

Programs

Maple

N:= 10^4: # to get all terms <= N
PhiE:= map(numtheory:-phi, [seq(i,i=2..N,2)]):
A:= NULL:
for n from 1 to N by 2 do
t:= numtheory:-phi(n);
if 2*t < n and member(t, PhiE[1..(n-1)/2]) then A:= A,n fi;
od:
A; # Robert Israel, Jan 06 2017

PARI

is(m) = m%2 && #select(k -> !(k%2) && k < m, invphi(eulerphi(m))) > 0; \\ Amiram Eldar, Nov 21 2024, using Max Alekseyev's invphi.gp