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A097942
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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).
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14
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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
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OFFSET
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1,2
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COMMENTS
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If you inspect PhiAnsYldList after running the Mathematica program below, the zeros with even-numbered indices should correspond to the nontotients (A005277).
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LINKS
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EXAMPLE
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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}.
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MAPLE
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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));
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MATHEMATICA
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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]
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PROG
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(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
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)]
(PARI)
{ A097942_list(n) = local(L, m, i, r);
m = 0;
for(i=1, n,
r = numinvphi(i);
if(r > m, print1(i, ", "); m = r) );
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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