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A308378 Numbers k such that phi(2k+1) = phi(2k+2).

%I A308378 #52 Jul 17 2022 02:21:25
%S A308378 0,1,7,127,247,487,1312,1627,1852,2593,5857,6682,9157,11467,12772,
%T A308378 23107,24607,24667,28822,32767,82087,92317,99157,107887,143497,153697,
%U A308378 159637,194122,198742,207637,245767,284407,294703,343492,420127
%N A308378 Numbers k such that phi(2k+1) = phi(2k+2).
%C A308378 For n > 0, 2*a(n) + 1 is a term of A020884. This is because 2*a(n) + 1 is odd and every odd number is the difference of the squares of two consecutive numbers and hence are coprime.
%C A308378 For n > 0, (2*a(n) + 1) * (2*a(n) + 2) is a term of A024364. This is because (2*a(n) + 1) * (2*a(n) + 2) = 2*((a(n) + 1)^2 + (a(n) + 1) * a(n)) and gcd((a(n) + 1), a(n)) = 1.
%C A308378 For n > 0, a(n) is congruent to 1 or 4 mod 6.
%C A308378 2*a(n) + 1 is congruent to 1 or 3 mod 6 and is a term of A047241.
%C A308378 2*a(n) + 2 is congruent to 2 or 4 mod 6 and is a term of A047235.
%H A308378 Amiram Eldar, <a href="/A308378/b308378.txt">Table of n, a(n) for n = 1..5416</a> (calculated using the b-file at A001274)
%F A308378 a(n) = (A299535(n) - 2) / 2.
%e A308378 0 is a term because phi(1) = phi(2) = 1.
%e A308378 1 is a term because phi(3) = phi(4) = 2.
%e A308378 7 is a term because phi(15) = phi(16) = 8.
%t A308378 Select[Range[0, 9999], EulerPhi[2# + 1] == EulerPhi[2# + 2] &] (* _Alonso del Arte_, Jul 05 2019 *)
%t A308378 Select[(#-1)/2&/@SequencePosition[EulerPhi[Range[900000]],{x_,x_}][[All,1]],IntegerQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 24 2019 *)
%o A308378 (PARI) lista(nn) = for(n=0, nn, if(eulerphi(2*n+1) == eulerphi(2*n+2), print1(n, ", ")));
%o A308378 lista(430000)
%Y A308378 Cf. A000010, A004767, A020884, A024364, A047235, A047241, A299535.
%Y A308378 Subset of A047234.
%Y A308378 Subset of A001274.
%K A308378 nonn
%O A308378 1,3
%A A308378 _Torlach Rush_, May 24 2019