A265733 Number of n-digit primes whose digits include at least n-1 digits "1".

4, 8, 10, 9, 10, 13, 7, 16, 10, 11, 11, 13, 9, 11, 6, 7, 8, 16, 7, 11, 8, 9, 9, 10, 8, 12, 6, 13, 13, 21, 7, 12, 8, 7, 15, 16, 8, 9, 17, 9, 5, 22, 9, 15, 6, 12, 9, 20, 11, 13, 14, 11, 13, 12, 9, 15, 7, 4, 8, 12, 4, 11, 8, 15, 10, 17, 12, 12, 4, 9, 9, 14, 8, 14, 10, 7, 10, 14, 5, 14

Offset: 1

Author

Michael De Vlieger and Robert G. Wilson v, Dec 14 2015

Keywords

• nonn
• base

Comments

Inspired by A241100 and its conjecture by Chai Wah Wu of Dec 10 2015.

The average value of a(n) is 10.6, median is 10, and mode is 11 for the first 1000 terms.

The first occurrence of k>0: 433, 361, 229, 1, 41, 15, 7, 2, 4, 3, 10, 26, 6, 51, 35, 8, 39, 180, 84, 48, 30, 42, 306, 138, 948, 642, ..., .

0 < a(n) < 27 for n < 1551.

It appears that a(n)/(9*n + 0^(n-1)) has its maximum at n = 2. - Altug Alkan, Dec 16 2015

Examples

			a(1) = 4 because {2, 3, 5, 7} are primes;
a(2) = 8 because {11, 13, 17, 19, 31, 41, 61, 71} are two digit primes with at least one digit being 1;
a(3) = 10 because {101, 113, 131, 151, 181, 191, 211, 311, 811, 911} are three digit primes with at least two digit being 1, etc.
		

Links

• Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1550

Crossrefs

Cf. A209252, A241100.

Programs

Maple

A:= proc(n) local x,X,i,y;
  x:= (10^n-1)/9;
  y:= [x,seq(seq(x+10^i*y,y=1..8),i=0..n-1),seq(x - 10^i,i=0..n-2)];
  nops(select(isprime,y))
end proc:
map(A, [$1..100]); # Robert Israel, Dec 31 2015

Mathematica

f1[n_] := Block[{cnt = k = 0, r = (10^n - 1)/9, s = {-1, 1, 2, 3, 4, 5, 6, 7, 8}}, If[ PrimeQ@ r, cnt++]; If[ PrimeQ[(10^(n - 1) - 1)/9], cnt--]; While[k < n, p = Select[r + 10^k*s, PrimeQ]; cnt += Length@ p; k++]; cnt]; Array[f1, 70]

Formula

Number of n-digit primes of the form 111...d...111, where the number of ones is at least n-1 and d is any other decimal digit.