

A217109


Minimal number (in decimal representation) with n nonprime substrings in base9 representation (substrings with leading zeros are considered to be nonprime).


16



2, 1, 12, 9, 83, 84, 81, 748, 740, 731, 729, 6653, 6581, 6563, 6564, 6561, 59222, 59069, 59068, 59051, 59052, 59049, 531614, 531569, 531464, 531460, 531452, 531443, 531441, 4784122, 4783142, 4783147, 4783070, 4782989, 4782971, 4782972, 4782969, 43048283
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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} 9^j, where k:=floor((sqrt(8*n+1)1)/2), i:= nA000217(k). For n=0,1,2,3,... the m(n) in base9 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 base9 representation), m>=d, than b := p*9^(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) >= 9^floor((sqrt(8*n7)1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 9^(n1), n>0.
a(A000217(n)k) >= 9^(n1) + k, 0<=k<n, n>0.
a(A000217(n)k) = 9^(n1) + p, where p is the minimal number >= 0 such that 9^(n1) + p, has k prime substrings in base9 representation, 0<=k<n, n>0.


EXAMPLE

a(0) = 2, since 2 = 2_9 is the least number with zero nonprime substrings in base9 representation.
a(1) = 1, since 1 = 1_9 is the least number with 1 nonprime substring in base9 representation.
a(2) = 12, since 12 = 13_9 is the least number with 2 nonprime substrings in base9 representation (1 and 13).
a(3) = 9, since 9 = 10_9 is the least number with 3 nonprime substrings in base9 representation (0, 1 and 10).
a(4) = 83, since 83 = 102_9 is the least number with 4 nonprime substrings in base9 representation, these are 0, 1, 10, and 02 (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: A135256 A090586 A268512 * A297967 A199930 A278330
Adjacent sequences: A217106 A217107 A217108 * A217110 A217111 A217112


KEYWORD

nonn,base


AUTHOR

Hieronymus Fischer, Dec 12 2012


STATUS

approved



