A266149 - OEIS

%I #21 Apr 22 2021 03:44:13

%S 4,21,46,43,40,53,35,49,40,38,44,52,35,45,49,42,38,57,28,45,38,47,38,

%T 52,33,45,56,38,36,65,29,56,48,40,38,58,37,33,57,40,37,61,41,39,37,44,

%U 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

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

%C 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.

%H Michael De Vlieger and Robert G. Wilson v, <a href="/A266149/b266149.txt">Table of n, a(n) for n = 1..1215</a>

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

%e a(1) = 4 since 2, 3, 5 and 7 are primes,

%e 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,

%e 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,

%e 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.

%t 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 *)

%o (Python)

%o from sympy import isprime

%o def a(n):

%o   if n == 1: return 4

%o   okset = set()

%o   for digit1 in "24568":

%o     for digit2 in "1379":

%o       t = int(digit1*(n-1) + digit2)

%o       if isprime(t): okset.add(t)

%o   for digit1 in "1379":

%o     for digit2 in "0123456789":

%o       if ((n-1)*int(digit1) + int(digit2))%3 == 0: continue

%o       for j in range(n):

%o         mc = digit1*j + digit2 + digit1*(n-1-j)

%o         if mc[0] == '0': continue

%o         t = int(mc)

%o         if isprime(t): okset.add(t)

%o   return len(okset)

%o print([a(n) for n in range(1, 76)]) # _Michael S. Branicky_, Apr 21 2021

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

%K base,nonn

%O 1,1

%A _Michael De Vlieger_ and _Robert G. Wilson v_, Dec 21 2015