cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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

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

A377727 Number of digit patterns of length n that satisfy no divisibility rules but do not generate primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 32, 9, 207
Offset: 1

Views

Author

Dmytro Inosov, Nov 05 2024

Keywords

Comments

Digit patterns (or digital types) are as per A266946.
The divisibility rules are per A376918 and they act to exclude patterns which always result in composite numbers, just due to the pattern.
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).
We call these digital types primonumerophobic and a(n) is the number of these of length n.
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.

Examples

			For n=10, the a(10) = 3 primonumerophobic patterns of length 10, which are also the smallest which exist, are
    pattern        A266946
   ----------     ----------
   AAABBBAAAB     1110001110
   AABABBBBBA     1101000001
   ABAAAAABBB     1011111000
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.

Links

Crossrefs

Cf. A267013, A376918, A164864, A266991

Formula

a(n) = A376918(n) - A267013(n).

Extensions

a(13) = 207 confirmed by Dmytro Inosov, Dec 23 2024
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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