cp's OEIS Frontend

A267013 Number of distinct digital types of n-digit primes in base 10.

%I A267013 #89 Jan 20 2025 16:23:38
%S A267013 1,2,4,11,51,177,876,3965,20782,114459,678536,4160910,27640731
%N A267013 Number of distinct digital types of n-digit primes in base 10.
%C A267013 The sequence is related to A266991.
%C A267013 Sequence {A164864(n) - a(n)}_(n>=1) begins 0,0,1,4,1,26,1,175,365,1516,...
%C A267013 One can explain, why, for example, a(4)=11, instead of A164864(4)=15. There exist exactly 4 types of 4-digit numbers, which cannot be prime. In A266946 these types are: 1001, 1010, 1100, 1111. Indeed, numbers abba,aabb,aaaa are divisible by 11; a number abab is divisible by 101.
%C A267013 In other cases of n-digit types we should verify the divisibility of numbers of types in A266946 at least by primes of the form 11,101,... Besides, a digital type 1...1 exists only for n in A004023, i.e., for  only 9 values of n from the first 270343. This simplifies the calculations.
%C A267013 a(n) <= A376918(n) with equality for n <= 9, but thereafter some digital types which pass the divisibility rules of A376918 don't in fact occur among the primes (see A377727). - _Dmytro Inosov_, Nov 05 2024
%C A267013 Based on the conjectured terms in A377727, the next three terms can be conjectured: a(14)=190402538; a(15)=1378294708; a(16)=10437142874. - _Dmytro Inosov_, Jan 07 2025
%H A267013 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. See pp. 14, 16-18.
%F A267013 a(n) = A376918(n) - A377727(n). - _Dmytro Inosov_, Nov 05 2024
%Y A267013 Cf. A164864, A164904, A264406, A266946, A376918, A377727.
%K A267013 nonn,base,more
%O A267013 1,2
%A A267013 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jan 08 2016
%E A267013 a(11)-a(13) from _Michael S. Branicky_, Nov 04 2024