

A217119


Greatest number (in decimal representation) with n nonprime substrings in base9 representation (substrings with leading zeros are considered to be nonprime).


9



47, 428, 1721, 6473, 14033, 35201, 58961, 58967, 465743, 530701, 530710, 1733741, 4250788, 4723108, 4776398, 25051529, 37327196, 42450640, 42986860, 42987589, 42996409, 225463817, 382055767, 382571822, 386888308, 386888419, 387356789
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OFFSET

0,1


COMMENTS

The sequence is welldefined 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} 9^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 base9 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (ki+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)k1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 3digit base9 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 < 9^(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..40


FORMULA

a(n) >= A217109(n).
a(n) >= A217309(A000217(num_digits_9(a(n)))n), where num_digits_9(x)=floor(log_9(x))+1 is the number of digits of the base9 representation of x.
a(n) <= 9^(n+2).
a(n) <= 9^min(n+2, 6*floor((n+7)/8)).
a(n) <= 9^((3/4)*(n + 3)).
a(n+m+1) >= 9*a(n), where m := floor(log_9(a(n))) + 1.


EXAMPLE

a(0) = 47, since 47 = 52_9 (base9) is the greatest number with zero nonprime substrings in base9 representation.
a(1) = 428 = 525_9 has 1 nonprime substring in base9 representation (= 525_9). All the other base9 substrings (2, 5, 5, 25, 52) are prime substrings. 525_9 is the greatest number with 1 nonprime substring.
a(2) = 1721 = 2322_9 has 10 substrings in base9 representation, exactly 2 of them are nonprime substrings (22_9 and 23_3=8), and there is no greater number with 2 nonprime substrings in base9 representation.
a(7) = 58967= 88788_9 has 15 substrings in base9 representation, exactly 7 of them are nonprime substrings (4times 8, 2times 88, and 8788), and there is no greater number with 7 nonprime substrings in base9 representation.


CROSSREFS

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300  A213321.
Cf. A217102  A217109, A217112  A217118.
Cf. A217302  A217309.
Sequence in context: A213216 A324625 A239539 * A142454 A200954 A126300
Adjacent sequences: A217116 A217117 A217118 * A217120 A217121 A217122


KEYWORD

nonn,base


AUTHOR

Hieronymus Fischer, Dec 20 2012


STATUS

approved



