OFFSET
0,1
COMMENTS
The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 7^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-7 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 3-digit base-7 number has at least 1 nonprime substring. Hence, each 3(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 7^(3n+2) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.
LINKS
Hieronymus Fischer, Table of n, a(n) for n = 0..50
FORMULA
a(n) >= A217107(n).
a(n) >= A217307(A000217(num_digits_7(a(n)))-n), where num_digits_7(x) is the number of digits of the base-7 representation of x.
a(n) <= 7^min(n+2, 5*floor((n+4)/5)).
a(n) <= 7^(n+2).
a(n) <= 7^min(3 + n/2, 8*floor((n+15)/16)).
a(n) <= 343*7^(n/2).
With m := floor(log_7(a(n))) + 1:
a(n+m+1) >= 7*a(n), if a(n)!=1 (mod 7).
a(n+m) >= 7*a(n), if a(n)=1 (mod 7).
EXAMPLE
a(0) = 37, since 37 = 52_7 (base-7) is the greatest number with zero nonprime substrings in base-7 representation.
a(1) = 331 = 652_7 has 1 nonprime substring in base-7 representation (= 6). All the other base-7 substrings (2, 5, 52_7=37, 65_7=47 and 652_7=331) are prime substrings. 331 is the greatest number with 1 nonprime substring.
a(2) = 317 = 632_7 has 6 substrings in base-7 representation (2, 3, 6, 32, 63 and 632), exactly 2 of them are nonprime substrings (6 and 32_6=20), and there is no greater number with 2 nonprime substrings in base-7 representation.
a(8) = 100291 = 565252_3 has 8 nonprime substrings in base-7 representation, these are 6, 252_7, 525_7, 565_7, 5252_7, 5652_7, 6525_7 and 65252_7. There is no greater number with 8 nonprime substrings in base-7 representation.
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Hieronymus Fischer, Dec 20 2012
STATUS
approved