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A217118 Greatest number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime). 2
491, 3933, 24303, 32603, 188143, 253789, 261117, 1555423, 2030319, 2088797, 2088943, 16185163, 16710383, 16710381, 16768991, 99606365, 129884143, 133683069, 134150015, 134209503, 770611067, 1039073149, 1069408239, 1073209071, 1073209083, 1073676029, 5065578363 (list; graph; refs; listen; history; text; internal format)
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} 8^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-8 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 4-digit base-8 number has at least 1 nonprime substring. Hence, each 4(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 8^(4n+3) 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) >= A217108(n).

a(n) >= A217308(A000217(num_digits_8(a(n)))-n), where num_digits_8(x) is the number of digits of the base-8 representation of x.

a(n) <= 8^min(n+3, 7*floor((n+6)/7)).

a(n) <= 512*8^n.

a(n+m+1) >= 8*a(n), where m := floor(log_8(a(n))) + 1.

EXAMPLE

a(0) = 491, since 491 = 753_8 (base-8) is the greatest number with zero nonprime substrings in base-8 representation.

a(1) = 3933 = 7535_8 has 1 nonprime substring in base-8 representation (=7535_8). All the other base-8 substrings are prime substrings. 3933 is the greatest such number with 1 nonprime substring.

a(2) = 24303 = 57357_8 has 15 substrings in base-8 representation, exactly 2 of them are nonprime substrings (57357_8 and 735_8), and there is no greater number with 2 nonprime substrings in base-3 representation.

a(3) = 32603 = 77533_8 has 15 substrings in base-8 representation, only 3 of them are nonprime substrings (33_8, 77_8, and 7753_8), and there is no greater number with 3 nonprime substrings in base-8 representation.

CROSSREFS

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079397, A213300 - A213321.

Cf. A217102 - A217109, A217112 - A217119.

Cf. A217302 - A217309.

Sequence in context: A271664 A082079 A260925 * A205200 A205058 A229520

Adjacent sequences:  A217115 A217116 A217117 * A217119 A217120 A217121

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Dec 20 2012

STATUS

approved

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Last modified July 1 11:09 EDT 2022. Contains 354972 sequences. (Running on oeis4.)