cp's OEIS Frontend

A175101 The number of bases b for which the odd squarefree semiprime A046388(n) is a Fermat pseudoprime.

%I A175101 #24 Mar 28 2021 07:03:25
%S A175101 2,2,2,2,2,2,2,2,14,2,2,14,2,34,2,2,2,2,2,2,2,34,2,2,14,2,2,2,2,2,14,
%T A175101 2,2,2,14,2,2,2,34,2,14,2,2,34,2,2,34,14,2,2,2,2,2,34,2,14,2,2,2,2,2,
%U A175101 2,2,2,98,2,14,2,14,2,2,2,2,34,2,2,2,2,2,34,2,14,2,98,2,34,2,2,142,14,2,14,2
%N A175101 The number of bases b for which the odd squarefree semiprime A046388(n) is a Fermat pseudoprime.
%C A175101 A number x is a Fermat pseudoprime for base b if b^(x-1) = 1 (mod x).
%C A175101 Comment from Karsten Meyer: (Start) Each term pq of the sequence A046388 is at least a Fermat pseudoprime to the two bases which have the property that |l*p - m*q| = 2 and b is the number between l*p and m*q. There are no more bases of this form below pq.
%C A175101 There may exist other bases smaller than pq, but just two bases have the property that they are direct neighbors of a multiple of p and a multiple of q. For example, 39=3*13 is a Fermat pseudoprime to the bases 14 and 25 because 14 is the number between 13 and 3*5 and 25 is the number between 3*8 and 2*13.
%C A175101 91=7*13 is a Fermat pseudoprime to the bases 27 and 64 because 27 is the number between 2*13 and 4*7 and 64 is the number between 9*7 and 5*13. For 91, the bases 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88 also exist, but neither of them lies between a multiple of 7 and a multiple of 13. (End)
%C A175101 Looking at odd squarefree semiprimes less than 10000, it appears that the number of bases is always of the form 2(2k^2-1), which is A060626 and twice A056220. Using the formula in A063994, the number of bases for pq (including bases 1 and pq-1) is gcd(p-1,pq-1) * gcd(q-1,pq-1).
%H A175101 Amiram Eldar, <a href="/A175101/b175101.txt">Table of n, a(n) for n = 1..10000</a>
%F A175101 a(n) = A063994(A046388(n)) - 2.
%e A175101 For A046388(1) = 15, the bases b in the range [2,13] are 4 and 11. So a(1) = 2.
%Y A175101 Cf. A046388, A063994 (number of bases b for which b^(n-1) = 1 (mod n)).
%K A175101 nonn
%O A175101 1,1
%A A175101 _T. D. Noe_, Dec 02 2010