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

Offset

1,10

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(13)=207, 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.

Links

Table of n, a(n) for n=1..12.

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

Formula

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

Example

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.

Crossrefs

Cf. A267013, A376918, A164864, A266991

Sequence in context: A210648 A119937 A254312 * A304048 A324223 A197368

Adjacent sequences: A377724 A377725 A377726 * A377728 A377729 A377730

Keyword

nonn,base,more

Author

Dmytro Inosov, Nov 05 2024

Status

approved