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A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

%I A262085 #42 Sep 08 2022 08:46:13
%S A262085 3,5,11,23,24,29,36,42,48,50,53,56,59,71,72,80,89,101,102,125,131,132,
%T A262085 149,173,176,191,230,233,248,263,269,359,368,389,401,431,449,464,479,
%U A262085 491,563,569,593,599,638,653,656,683,701,719,743,761,821,848,911,929,983
%N A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.
%C A262085 Sequence includes numbers n such that n and n + 8 are both prime (A023202).
%C A262085 Sequence also includes numbers n equal to 8*(a Mersenne prime) (cf A000668).
%C A262085 Sequence also includes n such that n/16 and n/8 + 1 are both odd primes.
%C A262085 Contains more composites than sequences A262084 and A262086. This is most likely due to the fact that 8 is a power of 2, as in A001838.
%H A262085 Robert Israel, <a href="/A262085/b262085.txt">Table of n, a(n) for n = 1..10000</a>
%H A262085 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>
%e A262085 3 since phi(11) = phi(3) + 8 (3 and 11 are both prime).
%e A262085 24 is a solution since phi(32) = phi(24) + 8 (24 is 8 * 3; 3 is a Mersenne prime).
%p A262085 select(t -> numtheory:-phi(t+8) = numtheory:-phi(t)+8, [$1..1000]); # _Robert Israel_, Mar 04 2016
%t A262085 Select[Range@1000, EulerPhi@(# + 8)== EulerPhi[#] + 8 &] (* _Vincenzo Librandi_, Sep 11 2015 *)
%o A262085 (Magma) [n: n in [1..1000] | EulerPhi(n+8) eq EulerPhi(n)+8]; // _Vincenzo Librandi_, Sep 11 2015
%o A262085 (PARI) is(n)=eulerphi(n + 8) == eulerphi(n) + 8 \\ _Anders Hellström_, Sep 11 2015
%o A262085 (Sage) [n for n in (1..1000) if euler_phi(n+8) == euler_phi(n)+8] # _Bruno Berselli_, Mar 04 2016
%Y A262085 Cf. A000010.
%Y A262085 Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262086 (k=10).
%K A262085 nonn,easy
%O A262085 1,1
%A A262085 _Kevin J. Gomez_, Sep 10 2015