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%I M1874 #64 Nov 27 2024 07:27:38
%S 1,2,8,48,80,96,128,240,288,480,1008,1200,1296,1440,1728,2592,2592,
%T 4800,5600,6480,8640,11040,12480,14976,19008,19200,22464,24320,24576,
%U 21120,28416,27840,25920,32000,32768,36000,47520,52992,60480,59904,79200,89280,96768
%N Values of phi(k) when phi(k) = phi(k+1).
%C In other words, consider k = 1,2,3,4,..., and if phi(k) = phi(k+1), add phi(k) to the sequence.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B36, pp. 138-142.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Amiram Eldar, <a href="/A003275/b003275.txt">Table of n, a(n) for n = 1..10755</a> (calculated from the b-file at A001274; terms 1..2567 from T. D. Noe)
%H Kathryn Miller, <a href="/A003275/a003275.pdf">The equation phi(n) = phi(n+1)</a>, Unpublished M.S., ND..
%H Kathryn Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [ See <a href="http://dx.doi.org/10.1090/S0025-5718-73-99703-2">Review</a>, Math. Comp., Vol. 27, p. 447-448, 1973 ].
%H Kathryn Miller, <a href="/A003275/a003275_1.pdf">Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000</a>, Mathematics of Computation 27 (1973), 47-48. (Annotated scanned copy)
%H Leo 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%F a(n) = A000010(A001274(n)). - _Reinhard Zumkeller_, May 20 2014
%t Cases[Split[Table[EulerPhi[k],{k,1,50000}]],{_,_}][[1;;27,1]] (* _Jean-François Alcover_, Mar 20 2011 *)
%t #[[1]]&/@Select[Partition[EulerPhi[Range[80000]],2,1],#[[1]]==#[[2]]&] (* _Harvey P. Dale_, Oct 03 2012 *)
%t SequenceCases[EulerPhi[Range[200000]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 05 2019 *)
%o (Haskell)
%o a003275 = a000010 . fromIntegral . a001274
%o -- _Reinhard Zumkeller_, May 20 2014
%o (PARI) lista(lim) = my(p1 = 1, p2); for(k = 2, lim, p2 = eulerphi(k); if(p1 == p2, print1(p1, ", ")); p1 = p2); \\ _Amiram Eldar_, Nov 27 2024
%Y Cf. A000010, A001274.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_