A217114 Greatest number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

11, 59, 239, 251, 751, 1007, 1019, 3823, 4079, 4055, 16111, 16087, 16319, 16367, 48991, 64351, 65263, 65269, 65471, 253919, 260959, 261079, 261847, 261871, 916319, 1043839, 1047391, 1044463, 1047511, 3665279, 3140991, 4189567, 4118519, 4177759, 4189565, 4193239, 14661117

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} 4^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-4 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-4 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 < 4^(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..100

Formula

a(n) >= A217104(n).

a(n) >= A217304(A000217(A110591(a(n)))-n).

a(n) <= 4^(n+2).

a(n) <= 4^min((n + 6)/2, 9*floor((n+18)/19)).

a(n) <= 64*4^(n/2).

a(n+m+1) >= 4*a(n), where m := floor(log_4(a(n))) + 1.

Example

a(0) = 11, since 11 = 23_4 (base-4) is the greatest number with zero nonprime substrings in base-4 representation.
a(1) = 59 = 323_4 has 6 substrings in base-4 representation (2, 3, 3, 23, 32 and 323), only 32_4=14 is a nonprime substring. 59 is the greatest such number with 1 nonprime substring.
a(2) = 239 = 3233_4 has 10 substrings in base-4 representation (2, 3, 3, 23, 32, 323, 233 and 3233), exactly 2 of them are nonprime substrings (32_4=14 and 33_4=15), and there is no greater number with 2 nonprime substrings in base-4 representation.
a(11) = 16087 = 3323113_4 has 28 substrings in base-4 representation. The base-4 nonprime substrings are 1, 1, 32, 33, 231, 332, 3113, 3231, 32311, 33321 and 323113. There is no greater number with 11 nonprime substrings in base-4 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: A186312 A142401 A139859 * A249891 A186256 A164299

Adjacent sequences: A217111 A217112 A217113 * A217115 A217116 A217117

Keyword

nonn,base

Author

Hieronymus Fischer, Dec 20 2012

Status

approved