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%I #44 Jan 06 2025 15:13:54
%S 0,0,0,0,0,0,0,0,0,3,32,9,207
%N Number of digit patterns of length n that satisfy no divisibility rules but do not generate primes.
%C Digit patterns (or digital types) are as per A266946.
%C The divisibility rules are per A376918 and they act to exclude patterns which always result in composite numbers, just due to the pattern.
%C There are A376918(n) remaining patterns but not all of them actually contain primes, and a(n) is how many of them do not, so that a(n) = A376918(n) - A267013(n).
%C We call these digital types primonumerophobic and a(n) is the number of these of length n.
%C It is conjectured that the next terms are a(14)=362, a(15)=363, a(16)=1448. This is based on the calculated number of primonumerophobic digit patterns with only 2 or 3 distinct digits and the vanishingly small combinatorial probability for the existence of additional primonumerophobic digit patterns of this length with 4 or more distinct digits.
%H Dmytro S. Inosov and Emil Vlasák, <a href="https://arxiv.org/abs/2410.21427">Cryptarithmically unique terms in integer sequences</a>, arXiv:2410.21427 [math.NT], 2024.
%F a(n) = A376918(n) - A267013(n).
%e For n=10, the a(10) = 3 primonumerophobic patterns of length 10, which are also the smallest which exist, are
%e pattern A266946
%e ---------- ----------
%e AAABBBAAAB 1110001110
%e AABABBBBBA 1101000001
%e ABAAAAABBB 1011111000
%e These patterns have 2 distinct digits (A and B) so that there are in total 81 numbers of each pattern that all happen to be composite despite the pattern coefficients in each having no common divisors.
%Y Cf. A267013, A376918, A164864, A266991
%K nonn,base,more
%O 1,10
%A _Dmytro Inosov_, Nov 05 2024
%E a(13) = 207 confirmed by _Dmytro Inosov_, Dec 23 2024