A081373 Number of values of k, 1 <= k <= n, with phi(k) = phi(n), where phi is Euler totient function, A000010.

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, 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, 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

Offset

1,2

Comments

Ordinal transform of Euler totient function phi, A000010. - Antti Karttunen, Aug 26 2024

Links

Paul Tek, Table of n, a(n) for n = 1..100000

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Example

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

Mathematica

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

Prog

(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
(PARI) a(n) = #select(x -> x <= n, invphi(eulerphi(n))); \\ Amiram Eldar, Nov 08 2024, using Max Alekseyev's invphi.gp

Crossrefs

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

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

Sequence in context: A072527 A345138 A343945 * A029436 A263274 A333940

Adjacent sequences: A081370 A081371 A081372 * A081374 A081375 A081376

Keyword

nonn

Author

Labos Elemer, Mar 24 2003

Status

approved