A217306 Minimal natural number (in decimal representation) with n prime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

1, 2, 11, 17, 47, 83, 269, 263, 479, 839, 1559, 1579, 2999, 5039, 9355, 9479, 14759, 56131, 56135, 61343, 56879, 336791, 341351, 336815, 341279, 341275, 2020727, 2020895, 2047651, 2020891, 4055159, 12098587, 12125347, 12285907, 15737755, 19128523, 39190247

Offset: 0

Author

Hieronymus Fischer, Nov 22 2012

Keywords

• nonn
• base

Comments

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=6*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 6^j = 2*(6^n - 1)/5 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-6) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-6 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 6.

Examples

			a(1) = 2 = 2_6, since 2 is the least number with 1 prime substring in base-6 representation.
a(2) = 11 = 15_6, since 11 is the least number with 2 prime substrings in base-6 representation (5_6=5 and 15_6=11).
a(3) = 17 = 25_6, since 17 is the least number with 3 prime substrings in base-6 representation (2_6, 5_6, and 25_6).
a(4) = 47 = 115_6, since 47 is the least number with 4 prime substrings in base-6 representation (5_6, 11_6=7, 15_6=11, and 115_6=47).
a(8) = 479 = 2115_6, since 479 is the least number with 8 prime substrings in base-6 representation (2_6, 5_6, 11_6=7, 15_6=11, 21_6=13, 115_6=47, 211_6=79, and 2115_6=479).
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..60

Crossrefs

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079397, A213300-A213321.

Cf. A217302-A217309.

Formula

a(n) > 6^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= 2*(6^n - 1)/5, n>0.

a(n+1) <= 6*a(n)+2.