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A293184 Numbers k such that bphi(k) = bphi(k+1), where bphi(k) is the bi-unitary analog of Euler's totient function (A116550).

%I A293184 #27 Jul 16 2022 07:14:16
%S A293184 1,14,20,57,187,188,916,1603,93928,142891,432976,549815,692259,773887,
%T A293184 872191,4297168,9478088,127162432,127991488,129015616,132527167
%N A293184 Numbers k such that bphi(k) = bphi(k+1), where bphi(k) is the bi-unitary analog of Euler's totient function (A116550).
%C A293184 187 is the first solution to bphi(k) = bphi(k+1) = bphi(k+2).
%C A293184 a(22) > 1.6*10^9, if it exists.  - _Amiram Eldar_, Jul 16 2022
%e A293184 14 is in the sequence since bphi(14) = bphi(15) = 9.
%t A293184 bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k];If[b1 == b2, a = AppendTo[a, k - 1]]; b1 = b2, {k, 1, 10^3}]; a (* after _Jean-François Alcover_ at A116550 *)
%o A293184 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A293184 gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
%o A293184 biuphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
%o A293184 isok(n) = biuphi(n) == biuphi(n+1);
%o A293184 lista(nn) = {x = biuphi(1); for (n=2, nn, y = biuphi(n); if (x==y, print1(n-1, ", ")); x = y;);} \\ _Michel Marcus_, Nov 09 2017
%Y A293184 Cf. A001274, A116550, A287055, A294030.
%K A293184 nonn,more
%O A293184 1,2
%A A293184 _Amiram Eldar_, Oct 01 2017
%E A293184 a(10) from _Michel Marcus_, Nov 11 2017
%E A293184 a(11) from _Michel Marcus_, Nov 12 2017
%E A293184 a(12)-a(21) from _Amiram Eldar_, Jul 16 2022