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

0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 32, 9, 207

Offset: 1

Author

Dmytro Inosov, Nov 05 2024

Keywords

• nonn
• base
• more

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

• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024.

Crossrefs

Cf. A267013, A376918, A164864, A266991

Formula

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

Extensions

a(13) = 207 confirmed by Dmytro Inosov, Dec 23 2024