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A179186 Numbers k such that phi(k) = phi(k+4), with Euler's totient function phi = A000010.

%I A179186 #77 Jul 28 2025 03:21:16
%S A179186 8,14,16,20,35,52,64,91,140,148,244,292,403,455,616,628,772,801,1011,
%T A179186 1024,1108,1144,1252,1270,1295,1456,1588,1684,1820,1828,2030,2164,
%U A179186 2452,2623,2644,2660,2692,2932,3028,3216,3321,3508,3988,4264,4340,4372,4612,4804,4852,4948
%N A179186 Numbers k such that phi(k) = phi(k+4), with Euler's totient function phi = A000010.
%C A179186 Is there some k > 5 such that phi(k) = phi(k+3)?
%C A179186 None up to 500000. - _Harvey P. Dale_, Feb 16 2011
%C A179186 No further solutions to the phi(k) = phi(k+3) problem less than 10^12. On the other hand, this sequence has 267797240 terms under 10^12. - _Jud McCranie_, Feb 13 2012
%C A179186 No reason is known that would prevent other solutions of phi(k) = phi(k+3), see Graham, Holt, & Pomerance. - _Jud McCranie_, Jan 03 2013
%C A179186 If a(n) is even then a(n)/2 is in A001494 - see comment at A217139. - _Jud McCranie_, Dec 31 2012
%D A179186 S. W. Graham, J. J. Holt, and C. Pomerance, "On the solutions to phi(n)=phi(n+k)", Number Theory in Progress, Proc. Intern. Conf. in Honor of 60th Birthday of A. Schinzel, Poland, 1997.  Walter de Gruyter, 1999, pp. 867-82.
%H A179186 Jud McCranie, <a href="/A179186/b179186.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A179186 F. Firoozbakht, <a href="http://www.primepuzzles.net/puzzles/puzz_466.htm">Puzzle 466: phi(n-1)=phi(n)=phi(n+1)</a>, in C. Rivera's Primepuzzles.
%H A179186 Kevin Ford, <a href="https://arxiv.org/abs/2002.12155">Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k)</a>, arXiv:2002.12155 [math.NT], 2020.
%t A179186 Select[Range[5000],EulerPhi[#]==EulerPhi[#+4]&]  (* _Harvey P. Dale_, Feb 16 2011 *)
%t A179186 SequencePosition[EulerPhi[Range[5000]],{x_,_,_,_,x_}][[;;,1]] (* _Harvey P. Dale_, Sep 12 2024 *)
%o A179186 (PARI) {op=vector(N=4); for( n=1,1e4,if( op[n%N+1]+0==op[n%N+1]=eulerphi(n),print1(n-N,",")))}
%o A179186 (Magma) [n: n in [1..5000] | EulerPhi(n) eq EulerPhi(n+4)]; // _Vincenzo Librandi_, Sep 08 2016
%Y A179186 Cf. A000010, A001274, A001494, A179187, A007015.
%K A179186 nonn
%O A179186 1,1
%A A179186 _M. F. Hasler_, Jan 05 2011