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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). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 1..86 (terms < 4*10^9, first 79 terms from T. D. Noe)

Wikipedia, Highly totient number

EXAMPLE

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}.

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]

PROG

(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

(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

CROSSREFS

A subsequence of A007374.

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

Sequence in context: A171647 A089821 A181808 * A004653 A115386 A058771

Adjacent sequences:  A097939 A097940 A097941 * A097943 A097944 A097945

KEYWORD

nonn

AUTHOR

Alonso del Arte, Sep 05 2004

EXTENSIONS

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

STATUS

approved

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Last modified October 30 07:54 EDT 2014. Contains 248796 sequences.