

A217107


Minimal number (in decimal representation) with n nonprime substrings in base7 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
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OFFSET

0,1


COMMENTS

The sequence is welldefined 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:= nA000217(k). For n=0,1,2,3,... the m(n) in base7 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (ki+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)k1+i = (k*(k+1)/2)+i = n, which proves the statement.
If p is a number with k prime substrings and d digits (in base7 representation), p != 1 (mod 7), m>=d, than b := p*7^(md) 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*n7)1)/2) for n>0, equality holds if n=1 or n+1 is a triangular number (cf. A000217).
a(A000217(n)1) = 7^(n1), n>1.
a(A000217(n)) = floor(400 * 7^(n4)), n>0.
a(A000217(n)) = 111…111_7 (with n digits), n>0.
a(A000217(n)k) >= 7^(n1) + k1, 1<=k<=n, n>1.
a(A000217(n)k) = 7^(n1) + p, where p is the minimal number >= 0 such that 7^(n1) + p, has k prime substrings in base7 representation, 1<=k<=n, n>1.


EXAMPLE

a(0) = 2, since 2 = 2_7 is the least number with zero nonprime substrings in base7 representation.
a(1) = 1, since 1 = 1_7 is the least number with 1 nonprime substring in base7 representation.
a(2) = 7, since 7 = 10_7 is the least number with 2 nonprime substrings in base7 representation (these are 0 and 1).
a(3) = 8, since 8 = 11_7 is the least number with 3 nonprime substrings in base7 representation (1, 1 and 11).
a(4) = 51, since 51 = 102_7 is the least number with 4 nonprime substrings in base7 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, A213300A213321.
Cf. A217102A217109.
Cf. A217302A217309.
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



