A081373 - OEIS

%I #33 Nov 08 2024 07:20:02

%S 1,2,1,2,1,3,1,2,2,3,1,4,1,3,1,2,1,4,1,3,2,2,1,4,1,3,2,4,1,5,1,2,2,3,

%T 1,5,1,3,2,4,1,6,1,3,3,2,1,5,2,4,1,4,1,4,2,5,2,2,1,6,1,2,3,2,1,5,1,3,

%U 1,6,1,7,1,4,3,5,2,8,1,4,1,4,1,9,1,3,1,5,1,10,2,2,3,2,3,5,1,4,4,6

%N Number of values of k, 1 <= k <= n, with phi(k) = phi(n), where phi is Euler totient function, A000010.

%C Ordinal transform of Euler totient function phi, A000010. - _Antti Karttunen_, Aug 26 2024

%H Paul Tek, <a href="/A081373/b081373.txt">Table of n, a(n) for n = 1..100000</a>

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%e For n = 16: phi(k) = {1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8} for k = 1,...,n; 2 numbers exist with phi(k) = phi(n) = 8: {15,16}, so a(16) = 2.

%e If n = p is an odd prime number, then a(p) = 1 with phi(k) = p-1.

%t f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] Table[f[w], {w, 1, 100}]

%o (PARI) a(n)=my(t=eulerphi(n), s); sum(k=1, n, eulerphi(k)==t) \\ _Charles R Greathouse IV_, Feb 21 2013, corrected by _Antti Karttunen_, Aug 26 2024

%o (PARI) a(n) = #select(x -> x <= n, invphi(eulerphi(n))); \\ _Amiram Eldar_, Nov 08 2024, using _Max Alekseyev_'s invphi.gp

%Y Cf. A000010, A081375 (positions of records), A210719 (of 1's).

%Y Cf. also A067004, A303756, A303757, A303777 (ordinal transform of this sequence).

%K nonn

%O 1,2

%A _Labos Elemer_, Mar 24 2003