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Numbers n such that phi(n+4) = phi(n) + 4.
6

%I #27 Sep 08 2022 08:45:01

%S 3,7,12,13,18,19,24,28,36,37,40,43,66,67,79,88,97,103,109,124,127,163,

%T 184,193,223,229,232,277,307,313,328,349,379,397,424,439,457,463,487,

%U 499,508,613,643,664,673,712,739,757,769,823,853,859,877,883,904,907

%N Numbers n such that phi(n+4) = phi(n) + 4.

%C In contrast with A015913, composite solutions are not rare. Prime solutions are common.

%C From _Kevin J. Gomez_, Mar 02 2016: (Start)

%C Composite solutions have two known forms:

%C n such that n = 4 * (2^p - 1) where 2^p - 1 is a Mersenne prime. (A001348)

%C n such that n = 8q where q is a Sophie Germain prime. (A005394)

%C There are composite solutions (such as 36) that do not fit either of these forms.

%C (End)

%H Seiichi Manyama, <a href="/A056772/b056772.txt">Table of n, a(n) for n = 1..1000</a>

%e n=1048: phi(1048)=520, phi(1048+4)=524.

%t Select[Range@1000, EulerPhi@(# + 4)== EulerPhi[#] + 4 &] (* _Vincenzo Librandi_, Sep 11 2015 *)

%t Position[Partition[EulerPhi[Range[1000]],5,1],_?(#[[1]]+4==#[[5]]&),1, Heads-> False]//Flatten (* _Harvey P. Dale_, Dec 18 2019 *)

%o (PARI) isok(n) = eulerphi(n+4) == eulerphi(n) + 4; \\ _Michel Marcus_, Sep 11 2015

%o (Magma) [n: n in [1..1000] | EulerPhi(n+4) eq EulerPhi(n)+4]; // _Vincenzo Librandi_, Sep 11 2015

%Y Cf. A015913 (sigma(n+4) = sigma(n) + 4).

%Y Cf. A001838 (k=2), this sequence (k=4), A262084 (k=6), A262085 (k=8), A262086 (k=10).

%K nonn

%O 1,1

%A _Labos Elemer_, Aug 17 2000