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

Author

Walter Nissen

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Corresponding values of n are given by A032447.

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

Programs

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

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 *)

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