A007614 All values attained by the phi(n) function, in ascending order.

1, 1, 2, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 8, 10, 10, 12, 12, 12, 12, 12, 12, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 20, 22, 22, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 28, 28, 30, 30, 32, 32, 32, 32, 32, 32, 32, 36, 36, 36, 36, 36, 36, 36, 36

Offset

1,3

Comments

Write down phi(1), phi(2), phi(3), ..., then sort this list. Of course the list before sorting is simply sequence A000010.

To ensure that all terms are found, the values of phi(n) should be computed for all n up to a primorial p# -- which are the local minima of the phi function. Selecting and sorting the values of phi(n) <= phi(p#) produces the terms of this sequence. - T. D. Noe, Mar 22 2011

A002202(n) occurs A058277(n) times. - Reinhard Zumkeller, Nov 22 2015

Links

Zak Seidov, Table of n, a(n) for n=1..9999 (values up to 5152)

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

Mathematica

Cases[Sort[Table[EulerPhi[n], {n, 1, 36^2}]], n_ /; n<=36 ]  (* Jean-François Alcover, Mar 22 2011 *)
A007614[m_]:=Select[Sort[Table[EulerPhi[n], {n, Prime[m]}]], #≤m&]; A007614[1000] (* Zak Seidov, Mar 22 2011 *)
primorial = Times @@ Prime[Range[4]]; phi = EulerPhi[primorial]; Sort[Select[EulerPhi[Range[primorial]], # <= phi &]] (* T. D. Noe, Mar 22 2011 *)

Prog

(PARI)  (See A032447).
(PARI) lista(nmax) = {my(m); for(n = 1, nmax, m = invphiNum(n); for(i = 1, m, print1(n, ", "))); } \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp
(Haskell)
import Data.List.Ordered (insertBag)
a007614 n = a007614_list !! (n-1)
a007614_list = f [1..] a002110_list [] where
   f xs'@(x:xs) ps'@(p:ps) us
     | x < p = f xs ps' $ insertBag (a000010' x) us
     | otherwise = vs ++ f xs' ps ws
     where (vs, ws) = span (<= a000010' x) us
-- Reinhard Zumkeller, Nov 22 2015

Crossrefs

Corresponding values of n are given by A032447.

Cf. A000010. A002110, A002202, A058277 (run lengths).

Sequence in context: A372057 A340743 A317841 * A289256 A287155 A287654

Adjacent sequences: A007611 A007612 A007613 * A007615 A007616 A007617

Keyword

nonn,easy,nice

Author

Walter Nissen

Status

approved