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A378507 The smallest number k such that the equation phi(phi(x)) = k has exactly n solutions.

%I A378507 #13 Apr 26 2025 06:00:29
%S A378507 10,56,6,1,84,312,2,200,464,36,108,4,12,88,816,264,440,360,552,120,
%T A378507 224,8,3696,1320,928,176,624,1472,832,5728,24,4560,1080,2000,16,2848,
%U A378507 72,1312,1872,80,1120,216,880,336,23360,448,3808,10608,648,528,352,9280,32
%N A378507 The smallest number k such that the equation phi(phi(x)) = k has exactly n solutions.
%C A378507 The smallest number k such that A378506(k) = n.
%C A378507 If phi(phi(x)) = k has a solution, then according to Carmichael's totient function conjecture there is at least one another number y != x such that phi(y) = phi(x) and then y is also a solution. Therefore, according to this conjecture, a(1) does not exist.
%H A378507 Amiram Eldar, <a href="/A378507/b378507.txt">Table of n, a(n) for n = 2..1000</a>
%H A378507 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A378507 David M. Bressoud, <a href="https://www.davidbressoud.org/books-videos">A Course in Computational Number Theory (web page)</a>, <a href="https://drive.google.com/file/d/1UMesJ93mpdCabQvETMqOBH8eCR4JWA6X/view?usp=sharing">CNT.m</a>, Computational Number Theory Mathematica package.
%H A378507 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function Conjecture</a>.
%H A378507 Wikipedia, <a href="https://en.wikipedia.org/wiki/Carmichael%27s_totient_function_conjecture">Carmichael's totient function conjecture</a>.
%t A378507 s[n_] := Sum[PhiMultiplicity[k], {k, PhiInverse[n]}]; seq[len_] := Module[{v = Table[0, {len+1}], c = 0, k = 1, ns}, While[c < len, ns = s[k]; If[0 < ns <= len + 1 && v[[ns]] == 0, v[[ns]] = k; c++]; k++]; Rest[v]]; seq[30] (* using David M. Bressoud's CNT.m *)
%o A378507 (PARI) s(n) = vecsum(apply(x -> invphiNum(x), invphi(n))); \\ using _Max Alekseyev_'s invphi.gp
%o A378507 lista(len) = {my(v = vector(len+1), c = 0, k = 1, ns); while(c < len, ns = s(k); if(ns > 0 && ns <= len + 1 && v[ns] == 0, c++; v[ns] = k); k++); vecextract(v,"^1");}
%Y A378507 Cf. A000010, A010554, A007374, A378506.
%K A378507 nonn
%O A378507 2,1
%A A378507 _Amiram Eldar_, Nov 29 2024