OFFSET
0,2
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):=8*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 8^j = 2*(8^n - 1)/7 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-8) 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-8 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 8.
LINKS
Hieronymus Fischer, Table of n, a(n) for n = 0..45
FORMULA
a(n) > 8^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= 2*(8^n - 1)/7, n>0.
a(n+1) <= 8*a(n)+2.
EXAMPLE
a(1) = 2 = 2_8, since 2 is the least number with 1 prime substring in base-8 representation.
a(2) = 11 = 13_8, since 11 is the least number with 2 prime substrings in base-8 representation (3_8 and 13_8).
a(3) = 19 = 23_8, since 19 is the least number with 3 prime substrings in base-8 representation (2_8, 3_8, and 23_8).
a(4) = 83 = 123_8, since 83 is the least number with 4 prime substrings in base-8 representation (2_8, 3_8, 23_8=19, and 123_8=83).
a(8) = 751 = 1357_8, since 751 is the least number with 8 prime substrings in base-8 representation (3_8, 5_8, 7_8, 13_8=11, 35_8=29, 57_8=47, 357_8=239, and 1357_8=751).
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Hieronymus Fischer, Nov 22 2012
STATUS
approved