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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.)