A217303 Minimal natural number (in decimal representation) with n prime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

1, 2, 5, 11, 17, 23, 50, 104, 71, 152, 215, 395, 476, 701, 719, 1367, 1934, 1448, 4127, 4121, 4346, 5822, 12302, 12383, 17468, 25505, 32066, 39113, 51749, 91040, 111509, 110798, 117359, 157211, 332396, 334358, 465092, 333791, 819386, 865232, 1001375, 1396673

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):=3*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 3^j = 3^n - 1 or m(n)=1, 2, 22, 222, 2222, 22222, …,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-3 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 3.

Examples

			a(1) = 2 = 2_3, since 2 is the least number with 1 prime substring in base-3 representation.
a(2) = 5 = 12_3, since 5 is the least number with 2 prime substrings in base-3 representation (2_3 and 12_3).
a(3) = 11 = 102_3, since 11 is the least number with 3 prime substrings in base-3 representation (2_3, 10_3, and 102_3).
a(5) = 23 = 212_3, since 23 is the least number with 5 prime substrings in base-3 representation (2 times 2_3, 12_3=5, 21_3=19, and 212_3=23).
a(7) = 104 = 10212_3, since 104 is the least number with 7 prime substrings in base-3 representation (2 times 2_3, 10_3=3, 12_3=5, 21_3=19, 102_3=11, and 212_3=23).
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..110

Crossrefs

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

Cf. A035244, A079397, A213300-A213321.

Cf. A217302-A217309.

Formula

a(n) > 3^floor(sqrt(8*n+1)-1)/2), for n>1.

a(n) <= 3^n - 1.

a(n+1) <= 3a(n)+2.