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A217309 Minimal natural number (in decimal representation) with n prime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime). 25
1, 2, 11, 23, 101, 173, 902, 1562, 1559, 8120, 14032, 14033, 73082, 126290, 604523, 657743, 723269, 1136684, 5918933, 5972147, 10227787, 25051529, 53276231, 54333278, 92071913, 441753767, 479669051, 483743986, 828662228, 3971590751, 4315446629 (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):=9*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 9^j = (9^n - 1)/4 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-9) 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-9 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 a prime number.

No term is divisible by 9.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..32

FORMULA

a(n) > 9^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= (9^n - 1)/4, n>0.

a(n+1) <= 9*a(n)+3.

EXAMPLE

a(1) = 2 = 2_9, since 2 is the least number with 1 prime substring in base-9 representation.

a(2) = 11 = 12_9, since 11 is the least number with 2 prime substrings in base-9 representation (2_9 and 12_9).

a(3) = 23 = 25_9, since 23 is the least number with 3 prime substrings in base-9 representation (2_9, 3_9, and 23_9).

a(4) = 101 = 122_9, since 101 is the least number with 4 prime substrings in base-9 representation (2 times 2_9, 12_9=11, and 122_9=101).

a(7) = 1562 = 2125_9, since 1562 is the least number with 7 prime substrings in base-9 representation (2 times 2_9, 5_9, 12_9=11, 21_9=19, 25_9=23, and 212_9=173).

CROSSREFS

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

Cf. A035244, A079397, A213300-A213321.

Cf. A217302-A217308.

Sequence in context: A198277 A218046 A342185 * A115374 A078699 A291679

Adjacent sequences:  A217306 A217307 A217308 * A217310 A217311 A217312

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Nov 22 2012

STATUS

approved

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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)