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%I #13 Oct 24 2014 04:15:20
%S 1,2,16,17,115,121,509,821,3251,4721,5749,22760,25301,41673,142950,
%T 173819,291714,920561,1222716,2041709,4450031,8559017,9350687,
%U 14295199,31150219,50568439,71502954,100066398,218051538,353979075,500526787,702815371,1512442643
%N Minimal natural number (in decimal representation) with n prime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).
%C 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):=7*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 7^j = (7^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-7) 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-7 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.
%C No term is divisible by 7.
%H Hieronymus Fischer, <a href="/A217307/b217307.txt">Table of n, a(n) for n = 0..40</a>
%F a(n) > 7^floor(sqrt(8*n-7)-1)/2), for n>0.
%F a(n) <= (7^n - 1)/3, n>0.
%F a(n+1) <= 7*a(n) + 2.
%e a(1) = 2 = 2_7, since 2 is the least number with 1 prime substring in base-7 representation.
%e a(2) = 16 = 22_7, since 16 is the least number with 2 prime substrings in base-7 representation (2 times 2_7=2).
%e a(3) = 17 = 23_7, since 17 is the least number with 3 prime substrings in base-7 representation (2_7, 3_7, and 23_7).
%e a(5) = 121 = 232_7, since 121 is the least number with 5 prime substrings in base-7 representation (2 times 2_7, 3_7, 23_7=17, and 32_7=23).
%e a(6) = 509 = 1325_7, since 509 is the least number with 6 prime substrings in base-7 representation (2_7, 3_7, 5_7, 25_7=19, 32_7=23, and 1325_7=509).
%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y Cf. A035244, A079397, A213300-A213321.
%Y Cf. A217302-A217309.
%K nonn,base
%O 0,2
%A _Hieronymus Fischer_, Nov 22 2012
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