A266149 Number of n-digit primes that consist of at least n-1 copies of some decimal digit.

4, 21, 46, 43, 40, 53, 35, 49, 40, 38, 44, 52, 35, 45, 49, 42, 38, 57, 28, 45, 38, 47, 38, 52, 33, 45, 56, 38, 36, 65, 29, 56, 48, 40, 38, 58, 37, 33, 57, 40, 37, 61, 41, 39, 37, 44, 36, 55, 47, 43, 47, 43, 35, 62, 43, 46, 29, 35, 37, 56, 39, 41, 46, 48, 39, 74, 45, 34, 34, 35, 34, 67, 39, 45, 43

Offset: 1

Author

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

Keywords

• base
• nonn

Comments

The first n at which a(n)=k for k=1...80, or 0 if no such k exists with n < 701: 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 433, 141, 181, 847, 19, 31, 253, 357, 137, 25, 68, 7, 29, 37, 10, 44, 5, 43, 16, 4, 11, 14, 3, 22, 33, 8, 139, 82, 12, 6, 102, 48, 27, 18, 36, 270, 198, 42, 54, 498, 90, 30, 738, 72, 222, 192, 852, 84, 342, 0, 66, 0, 816, 264, 0, 288, 0.

Examples

			a(1) = 4 since 2, 3, 5 and 7 are primes,
a(2) = 21 since 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97 are primes,
a(3) = 46 since 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997 are all primes,
a(4) = 43 since 1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 2333, 2777, 2999, 3313, 3323, 3331, 3343, 3373, 3433, 3533, 3733, 3833, 4111, 4441, 4447, 4999, 5333, 5557, 6661, 7177, 7333, 7477, 7577, 7717, 7727, 7757, 7877, 8111, 8887, 8999, 9199, 9929 and 9949 are primes; etc.
		

Links

• Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1215

Crossrefs

Cf. A265733, A266141, A266142, A266143, A266144, A266145, A266146, A266147, A266148.

Programs

Mathematica

Length /@ Array[Function[n, Select[Union[Flatten[Function[k, Select[FromDigits /@ Flatten[Permutations[Flatten@ {Table[k, {n - 1}], #}] & /@ Range[0, 9], 1], PrimeQ]] /@ Range[1, 9]]], Function[m, IntegerLength@ m == n]]], 100] (* Michael De Vlieger, Jan 01 2016 *)

Python

from sympy import isprime
def a(n):
  if n == 1: return 4
  okset = set()
  for digit1 in "24568":
    for digit2 in "1379":
      t = int(digit1*(n-1) + digit2)
      if isprime(t): okset.add(t)
  for digit1 in "1379":
    for digit2 in "0123456789":
      if ((n-1)*int(digit1) + int(digit2))%3 == 0: continue
      for j in range(n):
        mc = digit1*j + digit2 + digit1*(n-1-j)
        if mc[0] == '0': continue
        t = int(mc)
        if isprime(t): okset.add(t)
  return len(okset)
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Apr 21 2021

Formula

a(n) = A265733(n) + A266141(n) + A266142(n) + A266143(n) + A266144(n) + A266145(n) + A266146(n) + A266147(n) + A266148(n) for n>2.