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):=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.
LINKS
Hieronymus Fischer, Table of n, a(n) for n = 0..53
FORMULA
a(n) > 5^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= (5^n - 1)/2, n>0.
EXAMPLE
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).
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Hieronymus Fischer, Nov 22 2012
STATUS
approved