A250201 - OEIS

%I #36 Jan 25 2024 12:19:01

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

%C Phi_n(b, b-1) = (b-1)^EulerPhi(n) * Phi_n(b/(b-1)).

%C 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 If b = 1, then Phi_n(b, b-1) = 1 for all n, and 1 is not prime, so all a(n) > 1.

%C a(n) = 2 if and only if n is in A072226.

%C n        Phi_n(a, b)

%C 1        a-b

%C 2        a+b

%C 3        a^2+ab+b^2

%C 4        a^2+b^2

%C 5        a^4+a^3*b+a^2*b^2+a*b^3+b^4

%C 6        a^2-ab+b^2

%C ...      ...

%C n        b^EulerPhi(n)*Phi_n(a/b)

%H Eric Chen, <a href="/A250201/b250201.txt">Table of n, a(n) for n = 2..490</a>

%e a(11) = 6 because Phi_11(b, b-1) is composite for b = 2, 3, 4, 5 and prime for b = 6.

%e a(37) = 40 because Phi_37(b, b-1) is composite for b = 2, 3, 4, ..., 39 and prime for b = 40.

%t Table[k = 2; While[!PrimeQ[(k-1)^EulerPhi(n)*Cyclotomic[n, k/(k-1)]], k++]; k, {n, 2, 300}]

%o (PARI) a(n) = for(k = 2, 2^16, if(ispseudoprime((k-1)^eulerphi(n) * polcyclo(n, k/(k-1))), return(k)))

%Y Cf. A103794, A253633, A085398, A058013.

%K nonn

%O 2,1

%A _Eric Chen_, Mar 09 2015