

A217139


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


12



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
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OFFSET

1,1


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.


MATHEMATICA

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


PROG

(PARI) {op=vector(N=12); for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), print1(nN, ", ")))}
(MAGMA) [n: n in [1..3000]  EulerPhi(n) eq EulerPhi(n+12)]; // Vincenzo Librandi, Sep 08 2016


CROSSREFS

Cf. A000010, A179188, A217141, A007015.
Sequence in context: A205188 A333672 A045072 * A241481 A100409 A260921
Adjacent sequences: A217136 A217137 A217138 * A217140 A217141 A217142


KEYWORD

nonn,easy


AUTHOR

Michel Marcus, Sep 27 2012


STATUS

approved



