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A234399 a(n) = |{0 < k < n: 2^k*(2^phi(n-k) - 1) + 1 is prime}|, where phi(.) is Euler's totient function.

%I A234399 #18 Nov 05 2024 05:39:10
%S A234399 0,1,2,2,3,3,2,3,5,4,4,5,3,6,5,3,6,8,4,5,5,6,4,6,7,4,5,6,4,3,4,9,5,3,
%T A234399 8,5,4,3,8,8,3,8,6,7,7,8,8,9,4,5,8,9,7,6,10,11,4,6,6,8,8,10,4,4,7,4,
%U A234399 12,8,6,4,9,7,4,6,10,9,8,7,7,7,5,4,10,5,6,7,9,15,7,8,10,7,4,8,6,10,3,3,10,11
%N A234399 a(n) = |{0 < k < n: 2^k*(2^phi(n-k) - 1) + 1 is prime}|, where phi(.) is Euler's totient function.
%C A234399 Conjecture: a(n) > 0 for all n > 1.
%C A234399 See also the conjecture in A234388.
%C A234399 The conjecture is false. a(5962) = 0. - _Jason Yuen_, Nov 04 2024
%H A234399 Zhi-Wei Sun, <a href="/A234399/b234399.txt">Table of n, a(n) for n = 1..4000</a>
%e A234399 a(7) = 2 since 2^1*(2^phi(6)-1) + 1 = 2*3 + 1 = 7 and 2^2*(2^phi(5)-1) + 1 = 4*15 + 1 = 61 are both prime.
%t A234399 f[n_,k_]:=f[n,k]=2^k*(2^(EulerPhi[n-k])-1)+1
%t A234399 a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
%t A234399 Table[a[n],{n,1,100}]
%o A234399 (PARI) a(n) = sum(k=1,n-1,ispseudoprime(2^k*(2^eulerphi(n-k)-1)+1)) \\ _Jason Yuen_, Nov 04 2024
%Y A234399 Cf. A000010, A000040, A000079, A234309, A234310, A234337, A234344, A234346, A234347, A234359, A234360, A234361, A234388
%K A234399 nonn
%O A234399 1,3
%A A234399 _Zhi-Wei Sun_, Dec 25 2013