A217305 Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

1, 2, 7, 13, 37, 88, 67, 192, 317, 932, 942, 1567, 4663, 4692, 8442, 23317, 23442, 36067, 102217, 114192, 180337, 192317, 511087, 901682, 582942, 2495443, 2555436, 2536067, 5289942, 12321061, 12680337, 12301692, 26461592, 61508461, 61508462, 63885918

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):=5*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 5^j = (5^n - 1)/2 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-5) 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-5 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 a prime number.

No term is divisible by 5.

Examples

			a(1) = 2 = 2_5, since 2 is the least number with 1 prime substring in base-5 representation.
a(2) = 7 = 12_5, since 7 is the least number with 2 prime substrings in base-5 representation (2_5 and 12_5=7).
a(3) = 13 = 23_5, since 13 is the least number with 3 prime substrings in base-5 representation (2_5, 3_5, and 23_5).
a(4) = 37 = 122_5, since 37 is the least number with 4 prime substrings in base-5 representation (2 times 2_5, 12_5=7, and 122_5=37).
a(7) = 192 = 1232_5, since 192 is the least number with 7 prime substrings in base-5 representation (2 times 2_5, 3_5, 12_5=7, 23_5=13, 32_5=17, and 232_5=67).
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..53

Crossrefs

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217302-A217309.

Formula

a(n) > 5^floor(sqrt(8*n-7)-1)/2), for n>0.

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