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 A217303 Minimal natural number (in decimal representation) with n prime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime). 3
 1, 2, 5, 11, 17, 23, 50, 104, 71, 152, 215, 395, 476, 701, 719, 1367, 1934, 1448, 4127, 4121, 4346, 5822, 12302, 12383, 17468, 25505, 32066, 39113, 51749, 91040, 111509, 110798, 117359, 157211, 332396, 334358, 465092, 333791, 819386, 865232, 1001375, 1396673 (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):=3*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 3^j = 3^n - 1 or m(n)=1, 2, 22, 222, 2222, 22222, …,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-3 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 3. LINKS Hieronymus Fischer, Table of n, a(n) for n = 0..110 FORMULA a(n) > 3^floor(sqrt(8*n+1)-1)/2), for n>1. a(n) <= 3^n - 1. a(n+1) <= 3a(n)+2. EXAMPLE a(1) = 2 = 2_3, since 2 is the least number with 1 prime substring in base-3 representation. a(2) = 5 = 12_3, since 5 is the least number with 2 prime substrings in base-3 representation (2_3 and 12_3). a(3) = 11 = 102_3, since 11 is the least number with 3 prime substrings in base-3 representation (2_3, 10_3, and 102_3). a(5) = 23 = 212_3, since 23 is the least number with 5 prime substrings in base-3 representation (2 times 2_3, 12_3=5, 21_3=19, and 212_3=23). a(7) = 104 = 10212_3, since 104 is the least number with 7 prime substrings in base-3 representation (2 times 2_3, 10_3=3, 12_3=5, 21_3=19, 102_3=11, and 212_3=23). CROSSREFS Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685. Cf. A035244, A079397, A213300-A213321. Cf. A217302-A217309. Sequence in context: A078894 A086319 A220813 * A053033 A136244 A115057 Adjacent sequences: A217300 A217301 A217302 * A217304 A217305 A217306 KEYWORD nonn,base AUTHOR Hieronymus Fischer, Nov 22 2012 STATUS approved

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Last modified February 26 02:39 EST 2024. Contains 370335 sequences. (Running on oeis4.)