A217304 Minimal natural number (in decimal representation) with n prime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

1, 2, 7, 11, 23, 43, 93, 151, 239, 373, 479, 727, 1495, 2015, 2775, 5591, 6133, 7919, 12271, 22367, 24303, 30431, 48991, 89527, 95607, 98143, 129887, 357883, 358111, 382431, 744797, 519551, 1431007, 1432447, 1556319, 2457439

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):=4*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 4^j = 2*(4^n - 1)/3 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-4 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 4. a(1) = 2 is the only even term.

Examples

			a(1) = 2 = 2_4, since 2 is the least number with 1 prime substring in base-4 representation.
a(2) = 7 = 13_4, since 7 is the least number with 2 prime substrings in base-4 representation (3_4=3 and 13_4=7).
a(3) = 11 = 23_4, since 11 is the least number with 3 prime substrings in base-4 representation (2_4, 3_4, and 23_4).
a(5) = 43 = 223_4, since 43 is the least number with 5 prime substrings in base-4 representation (2 times 2_4, 3_4, 23_4=11, and 223_4=43).
a(7) = 151 = 2113_4, since 151 is the least number with 7 prime substrings in base-4 representation (2 times 2_4, 3_4, 11_4=5, 13_4=7, 113_4=23, and 2113_4=151).
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..77

Crossrefs

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

Cf. A035244, A079397, A213300-A213321.

Cf. A217302-A217309.

Formula

a(n) > 4^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= 2*(4^n - 1)/3, n>0.

a(n+1) <= 4*a(n) + 2.