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 A217107 Minimal number (in decimal representation) with n nonprime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime). 2
 2, 1, 7, 8, 51, 49, 57, 353, 345, 343, 400, 2417, 2411, 2403, 2401, 9604, 16880, 16823, 16829, 16809, 16807, 67228, 117763, 117721, 117666, 117659, 117651, 117649, 470596, 823709, 823664, 823615, 823560, 823553, 823545, 823543, 3294172, 5765310, 5765063 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 7^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-7 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement. If p is a number with k prime substrings and d digits (in base-7 representation), p != 1 (mod 7), m>=d, than b := p*7^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b. LINKS Hieronymus Fischer, Table of n, a(n) for n = 0..210 FORMULA a(n) >= 7^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n=1 or n+1 is a triangular number (cf. A000217). a(A000217(n)-1) = 7^(n-1), n>1. a(A000217(n)) = floor(400 * 7^(n-4)), n>0. a(A000217(n)) = 111…111_7 (with n digits), n>0. a(A000217(n)-k) >= 7^(n-1) + k-1, 1<=k<=n, n>1. a(A000217(n)-k) = 7^(n-1) + p, where p is the minimal number >= 0 such that 7^(n-1) + p, has k prime substrings in base-7 representation, 1<=k<=n, n>1. EXAMPLE a(0) = 2, since 2 = 2_7 is the least number with zero nonprime substrings in base-7 representation. a(1) = 1, since 1 = 1_7 is the least number with 1 nonprime substring in base-7 representation. a(2) = 7, since 7 = 10_7 is the least number with 2 nonprime substrings in base-7 representation (these are 0 and 1). a(3) = 8, since 8 = 11_7 is the least number with 3 nonprime substrings in base-7 representation (1, 1 and 11). a(4) = 51, since 51 = 102_7 is the least number with 4 nonprime substrings in base-7 representation, these are 0, 1, 02, and 102 (remember, that substrings with leading zeros are considered to be nonprime). CROSSREFS Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685. Cf. A035244, A079397, A213300-A213321. Cf. A217102-A217109. Cf. A217302-A217309. Sequence in context: A144749 A021463 A199964 * A320432 A141513 A258058 Adjacent sequences:  A217104 A217105 A217106 * A217108 A217109 A217110 KEYWORD nonn,base AUTHOR Hieronymus Fischer, Dec 12 2012 STATUS approved

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Last modified August 3 01:55 EDT 2021. Contains 346429 sequences. (Running on oeis4.)