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

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, 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, 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

Offset: 2

Author

Eric Chen, Mar 09 2015

Keywords

• nonn

Comments

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

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.

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

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

n Phi_n(a, b)

1 a-b

2 a+b

3 a^2+ab+b^2

4 a^2+b^2

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

6 a^2-ab+b^2

... ...

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

Examples

			a(11) = 6 because Phi_11(b, b-1) is composite for b = 2, 3, 4, 5 and prime for b = 6.
a(37) = 40 because Phi_37(b, b-1) is composite for b = 2, 3, 4, ..., 39 and prime for b = 40.
		

Links

• Eric Chen, Table of n, a(n) for n = 2..490

Crossrefs

Cf. A103794, A253633, A085398, A058013.

Programs

Mathematica

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

PARI

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