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A195418 a(n) = phi(C(n)) / gcd(C(n)-1, phi(C(n))), where C(n) is the n-th Cullen number. 1

%I #30 Dec 26 2023 09:43:06

%S 1,1,3,5,3,33,5,33,341,1045,189,1299,891,4437,9477,581,3855,105525,

%T 27825,23751,173043,10531345,56511,2386125,380955,256861,24926139,

%U 5108467,32397379,930343095,930291,36512775

%N a(n) = phi(C(n)) / gcd(C(n)-1, phi(C(n))), where C(n) is the n-th Cullen number.

%C When C(n) is prime (or 1), then a(n) = 1; that is, n is in A005849.

%C On the penultimate page of their paper, Grau and Luca ask for "a good (large) lower bound on this quantity which is valid for all n and which tends to infinity with n."

%H Amiram Eldar, <a href="/A195418/b195418.txt">Table of n, a(n) for n = 0..848</a>

%H José María Grau Ribas and Florian Luca, <a href="https://doi.org/10.1090/S0002-9939-2011-10899-2">Cullen numbers with the Lehmer property</a>, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134, <a href="http://arxiv.org/abs/1103.3578">preprint</a>, arXiv:1103.3578 [math.NT], Mar 18, 2011.

%e a(2) = 3 because the second Cullen number is 9; phi(9) = 6, therefore 6/gcd(8, 6) = 6/2 = 3.

%t cullen[n_] := n(2^n) + 1; Table[EulerPhi[cullen[n]]/GCD[cullen[n] - 1, EulerPhi[cullen[n]]], {n, 0, 39}]

%o (PARI) a(n)=my(C=n<<n,p=eulerphi(C+1));p/gcd(C,p) \\ _Charles R Greathouse IV_, Feb 05 2013

%Y Cf. A000010, A002064, A005849, A160595.

%K nonn,easy

%O 0,3

%A _Alonso del Arte_, Sep 20 2011

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Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)