A258915 The number of n-digit near-repdigit primes (A164937).

0, 0, 46, 43, 40, 53, 35, 49, 40, 38, 44, 52, 35, 45, 49, 42, 38, 57, 27, 45, 38, 47, 37, 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

Offset: 1

Author

Robert G. Wilson v, Jun 14 2015

Keywords

• nonn

Comments

The average is 44.25 with a standard deviation of about 9.48 for the first 1000 terms.

First occurrence of 20 < k < 80: 1132, ??22??, 1304, 433, 141, 181, 19, 118, 31, 253, 357, 137, 25, 68, 7, 29, 23, 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, ??73??, 66, ??75??, 816, 264, ??78??, 298; where ??xx?? denotes an unknown value for the index xx.

Roughly speaking, the probability that a random n-digit number is prime is about 1/(n*log(10)). The number of near-repdigit n-digit numbers is 81*n. Therefore it would be reasonable to expect around 81/log(10) (about 35) primes for each n. - Giovanni Resta, Jun 19 2015

Examples

			a(1) & a(2) = 0 by definition.
a(3) = 46 since there are 46 terms of 3 digits, see A164937(1) - A164937(46).
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1532

Crossrefs

Cf. A164937.

Essentially the same as A385280 but excluding near-repunit primes of A004022.

Programs

Mathematica

f[n_] := Block[{lst = {}, r = (10^(n - 1) - 1)/9}, Do[ AppendTo[ lst, DeleteCases[ Select[ FromDigits[ Permutations[ Append[ IntegerDigits[ a*r], d]]], PrimeQ@# && # > 100 &], r]], {a, 9}, {d, 0, 9}]; Length@ Union@ Flatten@ lst](* adapted after Arkadiusz Wesolowski of A164937 *) Array[f, 70]
(* to view the terms assign the terms in the b-file to "lst" and then *) ListPlot@ Sort@ lst (* and/or *) g[n_] := Count[lst, n]; DiscretePlot[ g[n], {n, 23, 80}]

Python

from gmpy2 import is_prime, digits
def a(n):
    if n < 3: return 0
    Rn = (10**n-1)//9
    return len(set(t for d in range(1, 10) for i in range(n if d in {1, 3, 7, 9} else 1) for c in set(range(-d, 10-d))-{0} if len(digits(t:=d*Rn+c*10**i))==n and is_prime(t)))
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Jun 28 2025