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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).
14
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
1,2
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
Jud McCranie, Table of n, a(n) for n = 1..109 (terms 1..79 from T. D. Noe, terms 80..86 from Donovan Johnson)
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.
Sequence in context: A326076 A181808 A343014 * A354541 A358513 A004653
KEYWORD
nonn
AUTHOR
Alonso del Arte, Sep 05 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, Sep 07 2004
STATUS
approved