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A217106 Minimal number (in decimal representation) with n nonprime substrings in base-6 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 (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} 6^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-6 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-6 representation), m>=d, than b := p*6^(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..300

FORMULA

a(n) >= 6^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).

a(A000217(n)) = 6^(n-1), n>0.

a(A000217(n)-k) >= 6^(n-1) + k, 0<=k<n, n>0.

a(A000217(n)-k) = 6^(n-1) + p, where p is the minimal number >= 0 such that 6^(n-1) + p, has k prime substrings in base-6 representation, 0<=k<n, n>0.

EXAMPLE

a(0) = 2, since 2 = 2_6 is the least number with zero nonprime substrings in base-6 representation.

a(1) = 1, since 1 = 1_6 is the least number with 1 nonprime substring in base-6 representation.

a(2) = 7, since 7 = 11_6 is the least number with 2 nonprime substrings in base-6 representation (1 and 1).

a(3) = 6, since 6 = 10_6 is the least number with 3 nonprime substrings in base-6 representation (0, 1 and 10).

a(4) = 41, since 41 = 105_6 is the least number with 4 nonprime substrings in base-6 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, A213300-A213321.

Cf. A217102-A217109.

Cf. A217302-A217309.

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

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Last modified April 12 11:49 EDT 2021. Contains 342920 sequences. (Running on oeis4.)