A217116 Greatest number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

23, 839, 5039, 6983, 7127, 42743, 45863, 46199, 275183, 279143, 277199, 1088531, 1674863, 1651103, 1674859, 6713711, 9906599, 10045559, 10072943, 39190247, 40278647, 60273359, 60295079, 60294239, 60437659, 241671887, 342609527, 359245007, 361640159, 362625959

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} 6^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-6 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-6 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 < 6^(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..55

Formula

a(n) >= A217106(n).

a(n) >= A217306(A000217(num_digits_6(a(n)))-n), where num_digits_6(x) is the number of digits of the base-6 representation of x.

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

a(n) <= 216*6^n.

a(n+m+1) >= 6*a(n), where m := floor(log_6(a(n))) + 1.

Example

(0) = 23, since 23 = 35_6 (base-6) is the greatest number with zero nonprime substrings in base-6 representation.
a(1) = 839 = 3515_6 has 1 nonprime substring in base-6 representation (= 1). All the other base-6 substrings (3, 5, 15_6=11, 35_6=23, 51_6=31, 351_6=139, 515_6=191 and 3515_6=839) are prime substrings. 839 is the greatest number with 1 nonprime substring.
a(2) = 5039 = 35155_6 has 2 nonprime substrings in base-6 representation (1 and 55_6=35), and there is no greater number with 2 nonprime substrings in base-6 representation.
a(5) = 42743 = 525515_6 has 5 nonprime substrings in base-6 representation, these are 1, 52_6=32, 55_6=35, 5515_6=1271 and 52551_6=7123, and there is no greater number with 5 nonprime substrings in base-6 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: A152183 A027410 A158505 * A081731 A335496 A183523

Adjacent sequences: A217113 A217114 A217115 * A217117 A217118 A217119

Keyword

base,nonn

Author

Hieronymus Fischer, Dec 20 2012

Status

approved