A097942 Highly totient numbers: each number k on this list has more solutions to the equation phi(x) = k than any preceding k (where phi is Euler's totient function, A000010).

1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 40320, 51840, 60480, 69120, 80640, 103680, 120960, 161280, 181440, 207360, 241920, 362880, 483840, 725760, 967680

Offset: 1

Author

Alonso del Arte, Sep 05 2004

Keywords

• nonn

Comments

If you inspect PhiAnsYldList after running the Mathematica program below, the zeros with even-numbered indices should correspond to the nontotients (A005277).

Where records occur in A014197. - T. D. Noe, Jun 13 2006

Cf. A131934.

Examples

			a(4) = 8 since phi(x) = 8 has the solutions {15, 16, 20, 24, 30}, one more solution than a(3) = 4 for which phi(x) = 4 has solutions {5, 8, 10, 12}.
		

Links

• Jud McCranie, Table of n, a(n) for n = 1..109 (terms 1..79 from T. D. Noe, terms 80..86 from Donovan Johnson)
• Wikipedia, Highly totient number.

Crossrefs

A subsequence of A007374.

Cf. A000010, A005277, A014573, A004653, A105207, A105208.

Programs

Maple

HighlyTotientNumbers := proc(n) # n > 1 is search maximum
local L, m, i, r; L := NULL; m := 0;
for i from 1 to n do
  r := nops(numtheory[invphi](i));
  if r > m then L := L,[i,r]; m := r fi
od; [L] end:
A097942_list := n -> seq(s[1], s = HighlyTotientNumbers(n));
A097942_list(500); # Peter Luschny, Sep 01 2012

Mathematica

searchMax = 2000; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; highlyTotientList = {1}; currHigh = 1; Do[If[phiAnsYldList[[n]] > phiAnsYldList[[currHigh]], highlyTotientList = {highlyTotientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyTotientList]

PARI

{ A097942_list(n) = local(L, m, i, r);
  m = 0;
  for(i=1, n,
\\ from Max Alekseyev, http://home.gwu.edu/~maxal/gpscripts/
   r = numinvphi(i);
   if(r > m, print1(i,", "); m = r) );
} \\ Peter Luschny, Sep 01 2012

Sage

def HighlyTotientNumbers(n) : # n > 1 is search maximum.
    R = {}
    for i in (1..n^2) :
        r = euler_phi(i)
        if r <= n :
            R[r] = R[r] + 1 if r in R else 1
    # print R.keys()   # A002202
    # print R.values() # A058277
    P = []; m = 1
    for l in sorted(R.keys()) :
        if R[l] > m : m = R[l]; P.append((l,m))
    # print [l[0] for l in P] # A097942
    # print [l[1] for l in P] # A131934
    return P
A097942_list = lambda n: [s[0] for s in HighlyTotientNumbers(n)]
A097942_list(500) # Peter Luschny, Sep 01 2012

Extensions

Edited and extended by Robert G. Wilson v, Sep 07 2004