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

8, 14, 16, 20, 35, 52, 64, 91, 140, 148, 244, 292, 403, 455, 616, 628, 772, 801, 1011, 1024, 1108, 1144, 1252, 1270, 1295, 1456, 1588, 1684, 1820, 1828, 2030, 2164, 2452, 2623, 2644, 2660, 2692, 2932, 3028, 3216, 3321, 3508, 3988, 4264, 4340, 4372, 4612, 4804, 4852, 4948

Offset: 1

Author

M. F. Hasler, Jan 05 2011

Keywords

• nonn

Comments

Is there some k > 5 such that phi(k) = phi(k+3)?

None up to 500000. - Harvey P. Dale, Feb 16 2011

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

No reason is known that would prevent other solutions of phi(k) = phi(k+3), see Graham, Holt, & Pomerance. - Jud McCranie, Jan 03 2013

If a(n) is even then a(n)/2 is in A001494 - see comment at A217139. - Jud McCranie, Dec 31 2012

References

• 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.

Links

• Jud McCranie, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• F. Firoozbakht, Puzzle 466: phi(n-1)=phi(n)=phi(n+1), in C. Rivera's Primepuzzles.
• Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Crossrefs

Cf. A000010, A001274, A001494, A179187, A007015.

Programs

Magma

[n: n in [1..5000] | EulerPhi(n) eq EulerPhi(n+4)]; // Vincenzo Librandi, Sep 08 2016

Mathematica

Select[Range[5000],EulerPhi[#]==EulerPhi[#+4]&]  (* Harvey P. Dale, Feb 16 2011 *)
SequencePosition[EulerPhi[Range[5000]],{x_,,,_,x_}][[;;,1]] (* Harvey P. Dale, Sep 12 2024 *)

PARI

{op=vector(N=4); for( n=1,1e4,if( op[n%N+1]+0==op[n%N+1]=eulerphi(n),print1(n-N,",")))}