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A217309
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Minimal natural number (in decimal representation) with n prime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).
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25
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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