A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

48, 68, 72, 78, 86, 88, 114, 143, 144, 156, 157, 164, 168, 186, 192, 203, 216, 222, 247, 273, 292, 356, 402, 432, 444, 450, 452, 456, 612, 654, 728, 732, 762, 798, 834, 864, 876, 884, 932, 942, 964, 1032, 1054, 1080, 1086, 1124, 1147, 1152, 1194, 1209, 1220

Offset: 1

Author

Michel Marcus, Sep 27 2012

Keywords

• nonn
• easy

Comments

Most of numbers n in this sequence are divisible by 2, and it appears that n/2 belongs to A179188. The other ones are listed in sequence A217141.

Proof of the comment: If n is even and not a multiple of 4 then phi(n)=phi(n/2). If n is a multiple of 4 then phi(n)=2 * phi(n/2). So when k is a multiple of 4 and phi(n)=phi(n+k), then phi(n/2)=phi(n/2+k/2). QED. This also applies to A179186, A179202. - Jud McCranie, Dec 30 2012

Links

• Jud McCranie, Table of n, a(n) for n = 1..10000
• Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Crossrefs

Cf. A000010, A179188, A217141, A007015.

Programs

Magma

[n: n in [1..3000] | EulerPhi(n) eq EulerPhi(n+12)]; // Vincenzo Librandi, Sep 08 2016

Mathematica

Select[Range[1, 5000], EulerPhi[#] == EulerPhi[# + 12] &] (* Vincenzo Librandi, Jun 24 2014 *)

PARI

{op=vector(N=12); for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), print1(n-N, ", ")))}