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A172314 Numbers k such that phi(k+1) = 4*phi(k).

%I A172314 #22 Sep 08 2022 08:45:50
%S A172314 1260,13650,17556,18720,24510,42120,113610,244530,266070,712080,
%T A172314 749910,795690,992250,1080720,1286730,1458270,1849470,2271060,2457690,
%U A172314 3295380,3370770,3414840,3714750,4061970,4736490,5314050,5827080,6566910,6935082,7303980,7864080
%N A172314 Numbers k such that phi(k+1) = 4*phi(k).
%D A172314 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
%D A172314 R. K. Guy, Unsolved Problems Number Theory, Sect. B36.
%H A172314 Donovan Johnson, <a href="/A172314/b172314.txt">Table of n, a(n) for n = 1..300</a>
%H A172314 V. L. Klee, Jr., <a href="http://www.jstor.org/stable/2305207">Some remarks on Euler's totient function</a>, Amer. Math. Monthly, 54 (1947), 332.
%H A172314 M. Lal and P. Gillard, <a href="http://www.jstor.org/stable/2005186">On the equation phi(n) = phi(n+k)</a>, Math. Comp. 26 (1972), 579-583.
%H A172314 K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. De Pauw University, 1972. [ Cf. <a href="http://www.jstor.org/stable/2005646">Review</a> on Math. Comp., Vol. 27, p. 447, 1973 ].
%H A172314 L. Moser, <a href="http://www.jstor.org/stable/2305815">Some equations involving Euler's totient function</a>, Amer. Math. Monthly, 56 (1949), 22-23.
%H A172314 A. Shinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa431.pdf">Sur l'équation phi(x+k) = phi(x)</a>, Acta Arith. 4 (1958), 181-184, [<a href="http://www.ams.org/mathscinet-getitem?mr=106867">MR0106857</a>]
%e A172314 phi(1260) = 288. phi(1261) = 1152. 4*phi(1260) = phi(1261).
%p A172314 with(numtheory): for n from 1 to 4000000 do; if 4*phi(n) = phi(n+1) then print(n); else fi ; od;
%t A172314 #[[1,1]]&/@Select[Partition[Table[{n,EulerPhi[n]},{n,4000000}],2,1], 4#[[1,2]]==#[[2,2]]&] (* _Harvey P. Dale_, Oct 11 2011 *)
%t A172314 Select[Range@1000000, EulerPhi@# 4 == EulerPhi[# + 1] &] (* _Vincenzo Librandi_, Jan 27 2016 *)
%o A172314 (Magma) [n: n in [1..2*10^6] | EulerPhi(n+1) eq 4*EulerPhi(n)]; // _Vincenzo Librandi_, Jan 27 2016
%Y A172314 Cf. A001274, A050472, A067143.
%K A172314 nonn
%O A172314 1,1
%A A172314 _Michel Lagneau_, Jan 31 2010
%E A172314 References separated by _R. J. Mathar_, Feb 19 2010