Search: a105208 -id:a105208
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A007374
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Smallest k such that phi(x) = k has exactly n solutions.
(Formerly M1093)
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+10
17
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1, 2, 4, 8, 12, 32, 36, 40, 24, 48, 160, 396, 2268, 704, 312, 72, 336, 216, 936, 144, 624, 1056, 1760, 360, 2560, 384, 288, 1320, 3696, 240, 768, 9000, 432, 7128, 4200, 480, 576, 1296, 1200, 15936, 3312, 3072, 3240, 864, 3120, 7344, 3888, 720, 1680, 4992
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OFFSET
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2,2
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COMMENTS
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The Carmichael Totient Conjecture is that there is no k such that phi(x) = k has a unique solution x. So a(1) does not exist.
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Wacław Sierpiński, Elementary Theory of Numbers, p. 234, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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MATHEMATICA
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a = Table[ 0, {10^5} ]; Do[ s = EulerPhi[ n ]; If[ s < 100001, a[ [ s ] ]++ ], {n, 1, 10^6} ]; Do[ k = 1; While[ a[ [ k ] ] != n, k++ ]; Print[ k ], {n, 2, 75} ]
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PROG
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(PARI) v=vectorsmall(10^6); for(n=1, 1e7, t=eulerphi(n); if(t<=#v, v[t]++))
u=vector(100); for(i=1, #v, t=v[i]; if(t&&t<=#u&&u[t]==0, u[t]=i)); u[2..#u]
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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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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+10
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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1, 2, 4, 8, 12, 32, 36, 40, 48, 160, 396, 2268, 2560, 3696, 9000, 15936, 17640, 22848, 29160, 38640, 81216, 91872, 153120, 225280, 228960, 410112, 494592, 540672, 619920, 900000, 1111968, 1282176, 1350720, 1932000, 2153088, 4093440, 5634720
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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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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