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

1,1

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.

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A273831 A273847 A306048 * A042223 A317225 A273780

Adjacent sequences: A266146 A266147 A266148 * A266150 A266151 A266152

Keyword

base,nonn

Author

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

Status

approved