

A217106


Minimal number (in decimal representation) with n nonprime substrings in base6 representation (substrings with leading zeros are considered to be nonprime).


2



2, 1, 7, 6, 41, 37, 36, 223, 224, 218, 216, 1319, 1307, 1301, 1297, 1296, 7829, 7793, 7787, 7783, 7778, 7776, 46703, 46709, 46679, 46673, 46663, 46658, 46656, 280205, 280075, 279983, 279979, 279949, 279941, 279938, 279936, 1679879, 1679807, 1679699, 1679669
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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} 6^j, where k:=floor((sqrt(8*n+1)1)/2), i:= nA000217(k). For n=0,1,2,3,... the m(n) in base6 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 base6 representation), m>=d, than b := p*6^(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..300


FORMULA

a(n) >= 6^floor((sqrt(8*n7)1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 6^(n1), n>0.
a(A000217(n)k) >= 6^(n1) + k, 0<=k<n, n>0.
a(A000217(n)k) = 6^(n1) + p, where p is the minimal number >= 0 such that 6^(n1) + p, has k prime substrings in base6 representation, 0<=k<n, n>0.


EXAMPLE

a(0) = 2, since 2 = 2_6 is the least number with zero nonprime substrings in base6 representation.
a(1) = 1, since 1 = 1_6 is the least number with 1 nonprime substring in base6 representation.
a(2) = 7, since 7 = 11_6 is the least number with 2 nonprime substrings in base6 representation (1 and 1).
a(3) = 6, since 6 = 10_6 is the least number with 3 nonprime substrings in base6 representation (0, 1 and 10).
a(4) = 41, since 41 = 105_6 is the least number with 4 nonprime substrings in base6 representation, these are 0, 1, 10, and 05 (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: A295850 A078104 A072280 * A329995 A086054 A256392
Adjacent sequences: A217103 A217104 A217105 * A217107 A217108 A217109


KEYWORD

nonn,base


AUTHOR

Hieronymus Fischer, Dec 12 2012


STATUS

approved



