A213306 -id:A213306

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A213307

Minimal prime with n nonprime substrings (Version 3: substrings with leading zeros are counted as nonprime if the corresponding number is not a prime).

+0
3

2, 13, 11, 127, 101, 149, 1009, 1063, 1049, 1481, 10091, 10069, 10169, 11681, 14669, 100129, 100189, 100169, 101681, 104681, 146669, 1000669, 1001219, 1001081, 1004669, 1014469, 1046849, 1468469, 10001081, 10004669, 10010851

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..100

FORMULA

a(n) > 10^floor((sqrt(8*n+1)-1)/2), for n>2.

a(n) >= A213304(n).

a(n) >= A213306(n).

EXAMPLE

a(0) = 2, since 2 is the least number with zero nonprime substrings.

a(1) = 13, since 13 there is one nonprime substring (=1).

a(2) = 11, since 11 is the least number with 2 nonprime substrings (2 times ‘1’).

a(3) = 127, since 127 is the least number with 3 nonprime substrings, these are 1 and 12 and 27 (according to version 3).

CROSSREFS

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

Cf. A035244, A079307, A213300 - A213321.

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Aug 26 2012

STATUS

approved

A213303

Smallest number with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

+0
3

2, 1, 10, 14, 101, 104, 144, 1001, 1014, 1044, 1444, 10010, 10014, 10144, 10444, 14444, 100101, 100104, 100144, 101444, 104444, 144444, 1000144, 1001014, 1001044, 1001444, 1014444, 1044444, 1444444, 10001044, 10001444, 10010144, 10010444, 10014444, 10144444, 10444444, 14444444, 100010144

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The sequence is well defined since for each n >= 0 there is a number with n nonprime substrings.

Different from A213304, first different term is a(16).

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..100

FORMULA

a(m(m+1)/2) = (13*10^(m-1)-4)/9, m>0.

With b(n):=floor((sqrt(8*n-7)-1)/2):

a(n) > 10^b(n), for n>2, a(n) = 10^b(n) for n=1,2.

a(n) >= 10^b(n)+4*10^(n-1-b(n)(b(n)+1)/2)-1)/9, equality holds if n or n+1 is a triangular number > 0 (cf. A000217).

a(n) <= A213304(n).

a(n) <= A213306(n).

EXAMPLE

a(0)=2, since 2 is the least number with zero nonprime substrings.

a(1)=1, since 1 has 1 nonprime substrings.

a(2)=10, since 10 is the least number with 2 nonprime substrings, these are 1 and 10 ('0' will not be counted).

a(3)=14, since 14 is the least number with 3 nonprime substrings, these are 1 and 4 and 14. 10, 11 and 12 only have 2 such substrings.