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Numbers n such that phi(n) = phi(n+8), with Euler's totient function phi=A000010.
7

%I #38 Sep 08 2022 08:45:54

%S 13,16,19,25,28,32,40,70,104,128,175,182,209,280,296,488,551,584,657,

%T 715,806,910,1232,1256,1544,1602,2022,2048,2216,2288,2504,2540,2590,

%U 2717,2912,3176,3368,3640,3656,4060,4328,4904,5246,5288,5320,5384,5864,5969

%N Numbers n such that phi(n) = phi(n+8), with Euler's totient function phi=A000010.

%C Among the 5596 terms below 10^7, a(6)=32 is the only term such that a(n+1) = a(n)+8.

%C There are 141741552 terms under 10^12. - _Jud McCranie_, Feb 13 2012

%C If a(n) is even then a(n)/2 is in A179186 - see comment at A217139. - _Jud McCranie_, Dec 31 2012

%H M. F. Hasler and Jud McCranie, <a href="/A179202/b179202.txt">Table of n, a(n) for n = 1..10000</a> (first 5596 terms from M. F. Hasler)

%H 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 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.

%F A000010(a(n)) = A000010(a(n)+8).

%t Select[Range[6000], EulerPhi[#] == EulerPhi[# + 8] &] (* _Vincenzo Librandi_, Sep 08 2016 *)

%o (PARI) {op=vector(N=8); for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), print1(n-N, ", ")))}

%o (Magma) [n: n in [1..10000] | EulerPhi(n) eq EulerPhi(n+8)]; // _Vincenzo Librandi_, Sep 08 2016

%Y Cf. A000010, A001274, A001494, A179186, A179187, A179188, A179189, A007015.

%K nonn

%O 1,1

%A _M. F. Hasler_, Jan 05 2011