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A250201 Least b such that Phi_n(b, b-1) is prime.

%I A250201 #36 Jan 25 2024 12:19:01
%S A250201 2,2,2,2,2,2,2,2,2,6,2,2,2,2,2,2,3,2,3,4,2,6,2,4,2,2,3,3,2,2,2,2,2,4,
%T A250201 5,40,2,3,2,7,2,5,3,3,2,13,3,2,14,4,22,3,3,13,2,34,5,3,5,2,2,34,9,2,
%U A250201 17,7,3,2,3,18,9,47,4,20,3,2,2,8,2,4,17,6,14,2,2,61,18,2,2
%N A250201 Least b such that Phi_n(b, b-1) is prime.
%C A250201 Phi_n(b, b-1) = (b-1)^EulerPhi(n) * Phi_n(b/(b-1)).
%C A250201 This sequence is not defined at n = 1 since Phi_1(b, b-1) = 1 for all b, and 1 is not prime. Conjecture: a(n) is defined for all n>1.
%C A250201 If b = 1, then Phi_n(b, b-1) = 1 for all n, and 1 is not prime, so all a(n) > 1.
%C A250201 a(n) = 2 if and only if n is in A072226.
%C A250201 n        Phi_n(a, b)
%C A250201 1        a-b
%C A250201 2        a+b
%C A250201 3        a^2+ab+b^2
%C A250201 4        a^2+b^2
%C A250201 5        a^4+a^3*b+a^2*b^2+a*b^3+b^4
%C A250201 6        a^2-ab+b^2
%C A250201 ...      ...
%C A250201 n        b^EulerPhi(n)*Phi_n(a/b)
%H A250201 Eric Chen, <a href="/A250201/b250201.txt">Table of n, a(n) for n = 2..490</a>
%e A250201 a(11) = 6 because Phi_11(b, b-1) is composite for b = 2, 3, 4, 5 and prime for b = 6.
%e A250201 a(37) = 40 because Phi_37(b, b-1) is composite for b = 2, 3, 4, ..., 39 and prime for b = 40.
%t A250201 Table[k = 2; While[!PrimeQ[(k-1)^EulerPhi(n)*Cyclotomic[n, k/(k-1)]], k++]; k, {n, 2, 300}]
%o A250201 (PARI) a(n) = for(k = 2, 2^16, if(ispseudoprime((k-1)^eulerphi(n) * polcyclo(n, k/(k-1))), return(k)))
%Y A250201 Cf. A103794, A253633, A085398, A058013.
%K A250201 nonn
%O A250201 2,1
%A A250201 _Eric Chen_, Mar 09 2015