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A217308 Minimal natural number (in decimal representation) with n prime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime). 3
1, 2, 11, 19, 83, 107, 157, 669, 751, 1259, 4957, 6879, 6011, 14303, 47071, 48093, 65371, 188143, 327515, 440287, 384751, 1029883, 2604783, 2948955, 3602299, 6946651, 20304733, 23846747, 23937003, 23723867, 57278299, 167689071, 175479547, 191496027, 233824091 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=8*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 8^j = 2*(8^n - 1)/7 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-8) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-8 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 8.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..45

FORMULA

a(n) > 8^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= 2*(8^n - 1)/7, n>0.

a(n+1) <= 8*a(n)+2.

EXAMPLE

a(1) = 2 = 2_8, since 2 is the least number with 1 prime substring in base-8 representation.

a(2) = 11 = 13_8, since 11 is the least number with 2 prime substrings in base-8 representation (3_8 and 13_8).

a(3) = 19 = 23_8, since 19 is the least number with 3 prime substrings in base-8 representation (2_8, 3_8, and 23_8).

a(4) = 83 = 123_8, since 83 is the least number with 4 prime substrings in base-8 representation (2_8, 3_8, 23_8=19, and 123_8=83).

a(8) = 751 = 1357_8, since 751 is the least number with 8 prime substrings in base-8 representation (3_8, 5_8, 7_8, 13_8=11, 35_8=29, 57_8=47, 357_8=239, and 1357_8=751).

CROSSREFS

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

Cf. A035244, A079397, A213300-A213321.

Cf. A217302-A217309.

Sequence in context: A186267 A067660 A235472 * A105076 A103200 A067670

Adjacent sequences:  A217305 A217306 A217307 * A217309 A217310 A217311

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Nov 22 2012

STATUS

approved

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Last modified April 17 23:03 EDT 2021. Contains 343071 sequences. (Running on oeis4.)