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A188466 Numbers n such that lambda(n) = lambda(n + lambda(n)).

%I A188466 #19 Sep 08 2022 08:45:56
%S A188466 1,4,6,16,36,55,78,105,124,144,171,200,253,325,406,465,666,689,715,
%T A188466 741,915,930,990,1027,1081,1136,1240,1421,1448,1610,1653,1711,1752,
%U A188466 1764,1800,1827,2211,2352,2448,2667,2800,2835,3403,3600,3619,3620,3660,3900,4840,4970,5253,5264,5513,5671,5886,6100,6328,8001,8112
%N A188466 Numbers n such that lambda(n) = lambda(n + lambda(n)).
%C A188466 Lambda is the function (A002322). If there are infinitely many Sophie Germain primes (conjecture), then this sequence is infinite. Proof: The numbers of the  form p(2p+1) are in a subsequence if p and 2p+1 are both prime with p > 3, because from the property that lambda(p(2p+1)) = p(p-1), if m = p(2p+1) then lambda(m+phi(m))  = lambda (p(2p+1) + p(p-1)) = lambda(3p^2) = p(p-1) = lambda(m).
%H A188466 Vincenzo Librandi, <a href="/A188466/b188466.txt">Table of n, a(n) for n = 1..1000</a>
%e A188466 36 is in the sequence because lambda(36) = 6, and lambda(36 + 6) = lambda(42) = 6.
%t A188466 Select[Range[20000], CarmichaelLambda[ #] == CarmichaelLambda[ # + CarmichaelLambda[#]  ] &]
%o A188466 (Magma) [1] cat [n: n in [2..8140] | CarmichaelLambda(n) eq CarmichaelLambda(n+CarmichaelLambda(n))];  // _Bruno Berselli_, Apr 10 2011
%o A188466 (PARI) lambda(n) = lcm(znstar(n)[2]);
%o A188466 isok(n) = lambda(n) == lambda(n+lambda(n)); \\ _Michel Marcus_, May 12 2018
%Y A188466 Cf. A002322, A005384.
%Y A188466 Cf. A185165: Numbers n such that lambda(n)= lambda(n - lambda(n)).
%Y A188466 Cf. A051487: Numbers n such that phi(n) = phi(n - phi(n)).
%Y A188466 Cf. A108569: Numbers n such that phi(n) = phi(n + phi(n)).
%K A188466 nonn
%O A188466 1,2
%A A188466 _Michel Lagneau_, Apr 01 2011