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A340325 Numbers k such that starting with k and repeatedly applying the map x -> A340323(x) reaches the fixed point 12.

%I A340325 #18 Feb 06 2021 12:35:37
%S A340325 5,6,10,11,12,13,14,15,17,18,19,20,22,23,24,25,26,28,29,30,33,34,35,
%T A340325 36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,
%U A340325 59,60,61,62,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,82,83,84
%N A340325 Numbers k such that starting with k and repeatedly applying the map x -> A340323(x) reaches the fixed point 12.
%C A340325 From _Sebastian Karlsson_, Jan 15 2021: (Start)
%C A340325 The sequence contains no powers of two. If a number isn't a power of two, then it is in this sequence if and only if either of the following conditions hold:
%C A340325 - It is a multiple of a prime that is not a Mersenne prime.
%C A340325 - It is divisible by the square of a Mersenne prime greater than 3. (End)
%t A340325 fa[n_]:=fa[n]=FactorInteger[n];phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
%t A340325 phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}];
%t A340325 S[n_] := NestWhile [phi, n, ! ( # == 12 || # == 3 || # == 4) &];
%t A340325 Select[1 + Range[100], S[#] == 12 &]
%o A340325 (PARI) f(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, (f[i, 1]+1)*((f[i, 1]-1)^(f[i, 2]-1)))); \\ A340323
%o A340325 isok(m) = if (m==1, return(0)); while(! ((m==3) || (m==4) || (m==12)), m = f(m)); (m==12); \\ _Michel Marcus_, Jan 21 2021
%Y A340325 Cf. A340323, A340324.
%Y A340325 Cf. A000668.
%K A340325 nonn
%O A340325 1,1
%A A340325 _José María Grau Ribas_, Jan 07 2021