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

1, 2, 4, 11, 51, 177, 876, 3965, 20782, 114459, 678536, 4160910, 27640731

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2016

Keywords

• nonn
• base
• more

Comments

The sequence is related to A266991.

Sequence {A164864(n) - a(n)}_(n>=1) begins 0,0,1,4,1,26,1,175,365,1516,...

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.

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.

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

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

Links

• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 14, 16-18.

Crossrefs

Cf. A164864, A164904, A264406, A266946, A376918, A377727.

Formula

a(n) = A376918(n) - A377727(n). - Dmytro Inosov, Nov 05 2024

Extensions

a(11)-a(13) from Michael S. Branicky, Nov 04 2024