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A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

%I A056772 #37 Mar 13 2025 08:56:11
%S A056772 3,7,12,13,18,19,24,28,36,37,40,43,66,67,79,88,97,103,109,124,127,163,
%T A056772 184,193,223,229,232,277,307,313,328,349,379,397,424,439,457,463,487,
%U A056772 499,508,613,643,664,673,712,739,757,769,823,853,859,877,883,904,907
%N A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.
%C A056772 In contrast with A015913, composite solutions are not rare. Prime solutions are common.
%C A056772 From _Kevin J. Gomez_, Mar 02 2016: (Start)
%C A056772 Composite solutions have two known forms:
%C A056772   n such that n = 4 * (2^p - 1) where 2^p - 1 is a Mersenne prime. (A001348)
%C A056772   n such that n = 8q where q is a Sophie Germain prime. (A005394)
%C A056772 There are composite solutions (such as 36) that do not fit either of these forms. (End)
%H A056772 Amiram Eldar, <a href="/A056772/b056772.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Seiichi Manyama)
%e A056772 For k = 1048: phi(1048) = 520, phi(1048+4) = 524.
%t A056772 Select[Range@1000, EulerPhi@(# + 4)== EulerPhi[#] + 4 &] (* _Vincenzo Librandi_, Sep 11 2015 *)
%t A056772 Position[Partition[EulerPhi[Range[1000]],5,1],_?(#[[1]]+4==#[[5]]&),1, Heads-> False]//Flatten (* _Harvey P. Dale_, Dec 18 2019 *)
%o A056772 (PARI) isok(n) = eulerphi(n+4) == eulerphi(n) + 4; \\ _Michel Marcus_, Sep 11 2015
%o A056772 (Magma) [n: n in [1..1000] | EulerPhi(n+4) eq EulerPhi(n)+4]; // _Vincenzo Librandi_, Sep 11 2015
%Y A056772 Cf. A000010, A015913 (sigma(k+4) = sigma(k) + 4).
%Y A056772 Cf. A001838 (k=2), this sequence (k=4), A262084 (k=6), A262085 (k=8), A262086 (k=10).
%K A056772 nonn
%O A056772 1,1
%A A056772 _Labos Elemer_, Aug 17 2000