A217302 -id:A217302 - cp's OEIS frontend

A217106 Minimal number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

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

Offset: 0

Hieronymus Fischer, Dec 12 2012

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.

			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).

Hieronymus Fischer, Table of n, a(n) for n = 0..300

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

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<=k0.
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<=k0.

A217107 Minimal number (in decimal representation) with n nonprime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Dec 12 2012

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.

			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).

Hieronymus Fischer, Table of n, a(n) for n = 0..210

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

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.

A217108 Minimal number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 10, 8, 67, 66, 64, 523, 525, 514, 512, 4127, 4115, 4099, 4098, 4096, 32797, 32799, 32779, 32771, 32770, 32768, 262237, 262239, 262173, 262163, 262147, 262146, 262144, 2097391, 2097259, 2097211, 2097181, 2097169, 2097163, 2097154, 2097152, 16777695

Offset: 0

Hieronymus Fischer, Dec 12 2012

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} 8^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-8 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-8 representation), m>=d, than b := p*8^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_8 is the least number with zero nonprime substrings in base-8 representation.
a(1) = 1, since 1 = 1_8 is the least number with 1 nonprime substring in base-8 representation.
a(2) = 10, since 10 = 12_8 is the least number with 2 nonprime substrings in base-8 representation (1 and 12).
a(3) = 8, since 8 = 10_8 is the least number with 3 nonprime substrings in base-8 representation (0, 1 and 10).
a(4) = 67, since 67 = 103_8 is the least number with 4 nonprime substrings in base-8 representation, these are 0, 1, 10, and 03 (remember, that substrings with leading zeros are considered to be nonprime).

Hieronymus Fischer, Table of n, a(n) for n = 0..210

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

a(n) >= 8^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 8^(n-1), n>0.
a(A000217(n)-k) >= 8^(n-1) + k, 0<=k0.
a(A000217(n)-k) = 8^(n-1) + p, where p is the minimal number >= 0 such that 8^(n-1) + p, has k prime substrings in base-8 representation, 0<=k0.

A217113 Greatest number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 23, 71, 26, 77, 233, 239, 719, 701, 647, 725, 2159, 2177, 2158, 2157, 5822, 5741, 6551, 6476, 6532, 6531, 18944, 19436, 19655, 19601, 19673, 19653, 58310, 58309, 58316, 58967, 59021, 58964, 157211, 157217, 174950, 176408, 176407, 176903, 177065, 177064, 471653, 511511

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 3^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-3 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. This proves the statement of existence. Proof of finiteness: Each 2-digit base-3 number has at least 1 nonprime substring. Hence, each 2(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 3^(2n+1) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 2, since 2 = 2_3 (base-3) is the greatest number with zero nonprime substrings in base-3 representation.
a(1) = 23 = 212_3 has 1 substring in base-3 representation (= 1). All the other base-3 substrings (2, 2, 21, 12, 212) are prime substrings. 23 is the greatest number with 1 nonprime substring.
a(2) = 71 = 2122_3 has 10 substrings in base-3 representation (1, 2, 2, 2, 12, 21, 22, 122, 212, 2122), exactly 2 of them are nonprime substrings (1 and 22_3=8), and there is no greater number with 2 nonprime substrings in base-3 representation.
a(3) = 26 = 222_3 has 6 substrings in base-3 representation, only 3 of them are prime substrings (2, 2, 2) which implies that exactly 3 substrings must be nonprime, and there is no greater number with 3 nonprime substrings in base-3 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..250

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217103(n).
a(n) >= A217303(A000217(A081604(a(n)))-n).
Example: a(12)=2177=2222122_3, A000217(A081604(2177))=28, hence a(12)>=A217303(28-12)=1934.
a(n) <= 3^min(n + 2, 5*floor((n+4)/5)).
a(n) <= 3^(n + 2).
a(n) <= 3^min((n + 11)/3, 11*floor((n+32)/33)).
a(n) <= 3^((1/3)*(n + 11)).
With m := floor(log_3(a(n))) + 1:
a(n+m+1) >= 3*a(n), if a(n)!=1 (mod 3).
a(n+m) >= 3*a(n), if a(n)=1 (mod 3).

A217118 Greatest number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

491, 3933, 24303, 32603, 188143, 253789, 261117, 1555423, 2030319, 2088797, 2088943, 16185163, 16710383, 16710381, 16768991, 99606365, 129884143, 133683069, 134150015, 134209503, 770611067, 1039073149, 1069408239, 1073209071, 1073209083, 1073676029, 5065578363

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 8^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-8 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. This proves the statement of existence. Proof of finiteness: Each 4-digit base-8 number has at least 1 nonprime substring. Hence, each 4(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 8^(4n+3) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 491, since 491 = 753_8 (base-8) is the greatest number with zero nonprime substrings in base-8 representation.
a(1) = 3933 = 7535_8 has 1 nonprime substring in base-8 representation (=7535_8). All the other base-8 substrings are prime substrings. 3933 is the greatest such number with 1 nonprime substring.
a(2) = 24303 = 57357_8 has 15 substrings in base-8 representation, exactly 2 of them are nonprime substrings (57357_8 and 735_8), and there is no greater number with 2 nonprime substrings in base-3 representation.
a(3) = 32603 = 77533_8 has 15 substrings in base-8 representation, only 3 of them are nonprime substrings (33_8, 77_8, and 7753_8), and there is no greater number with 3 nonprime substrings in base-8 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..50

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217108(n).
a(n) >= A217308(A000217(num_digits_8(a(n)))-n), where num_digits_8(x) is the number of digits of the base-8 representation of x.
a(n) <= 8^min(n+3, 7*floor((n+6)/7)).
a(n) <= 512*8^n.
a(n+m+1) >= 8*a(n), where m := floor(log_8(a(n))) + 1.

A217304 Minimal natural number (in decimal representation) with n prime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 7, 11, 23, 43, 93, 151, 239, 373, 479, 727, 1495, 2015, 2775, 5591, 6133, 7919, 12271, 22367, 24303, 30431, 48991, 89527, 95607, 98143, 129887, 357883, 358111, 382431, 744797, 519551, 1431007, 1432447, 1556319, 2457439

Offset: 0

Hieronymus Fischer, Nov 22 2012

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=4*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 4^j = 2*(4^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222, …,for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-4 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 4. a(1) = 2 is the only even term.

			a(1) = 2 = 2_4, since 2 is the least number with 1 prime substring in base-4 representation.
a(2) = 7 = 13_4, since 7 is the least number with 2 prime substrings in base-4 representation (3_4=3 and 13_4=7).
a(3) = 11 = 23_4, since 11 is the least number with 3 prime substrings in base-4 representation (2_4, 3_4, and 23_4).
a(5) = 43 = 223_4, since 43 is the least number with 5 prime substrings in base-4 representation (2 times 2_4, 3_4, 23_4=11, and 223_4=43).
a(7) = 151 = 2113_4, since 151 is the least number with 7 prime substrings in base-4 representation (2 times 2_4, 3_4, 11_4=5, 13_4=7, 113_4=23, and 2113_4=151).

Hieronymus Fischer, Table of n, a(n) for n = 0..77

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217302-A217309.

a(n) > 4^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= 2*(4^n - 1)/3, n>0.
a(n+1) <= 4*a(n) + 2.

A217305 Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 7, 13, 37, 88, 67, 192, 317, 932, 942, 1567, 4663, 4692, 8442, 23317, 23442, 36067, 102217, 114192, 180337, 192317, 511087, 901682, 582942, 2495443, 2555436, 2536067, 5289942, 12321061, 12680337, 12301692, 26461592, 61508461, 61508462, 63885918

Offset: 0

Hieronymus Fischer, Nov 22 2012

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=5*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 5^j = (5^n - 1)/2 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-5) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-5 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be a prime number.

No term is divisible by 5.

			a(1) = 2 = 2_5, since 2 is the least number with 1 prime substring in base-5 representation.
a(2) = 7 = 12_5, since 7 is the least number with 2 prime substrings in base-5 representation (2_5 and 12_5=7).
a(3) = 13 = 23_5, since 13 is the least number with 3 prime substrings in base-5 representation (2_5, 3_5, and 23_5).
a(4) = 37 = 122_5, since 37 is the least number with 4 prime substrings in base-5 representation (2 times 2_5, 12_5=7, and 122_5=37).
a(7) = 192 = 1232_5, since 192 is the least number with 7 prime substrings in base-5 representation (2 times 2_5, 3_5, 12_5=7, 23_5=13, 32_5=17, and 232_5=67).

Hieronymus Fischer, Table of n, a(n) for n = 0..53

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217302-A217309.

a(n) > 5^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= (5^n - 1)/2, n>0.

A217306 Minimal natural number (in decimal representation) with n prime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 11, 17, 47, 83, 269, 263, 479, 839, 1559, 1579, 2999, 5039, 9355, 9479, 14759, 56131, 56135, 61343, 56879, 336791, 341351, 336815, 341279, 341275, 2020727, 2020895, 2047651, 2020891, 4055159, 12098587, 12125347, 12285907, 15737755, 19128523, 39190247

Offset: 0

Hieronymus Fischer, Nov 22 2012

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=6*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 6^j = 2*(6^n - 1)/5 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-6) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-6 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 6.

			a(1) = 2 = 2_6, since 2 is the least number with 1 prime substring in base-6 representation.
a(2) = 11 = 15_6, since 11 is the least number with 2 prime substrings in base-6 representation (5_6=5 and 15_6=11).
a(3) = 17 = 25_6, since 17 is the least number with 3 prime substrings in base-6 representation (2_6, 5_6, and 25_6).
a(4) = 47 = 115_6, since 47 is the least number with 4 prime substrings in base-6 representation (5_6, 11_6=7, 15_6=11, and 115_6=47).
a(8) = 479 = 2115_6, since 479 is the least number with 8 prime substrings in base-6 representation (2_6, 5_6, 11_6=7, 15_6=11, 21_6=13, 115_6=47, 211_6=79, and 2115_6=479).

Hieronymus Fischer, Table of n, a(n) for n = 0..60

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217302-A217309.

a(n) > 6^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= 2*(6^n - 1)/5, n>0.
a(n+1) <= 6*a(n)+2.

A217307 Minimal natural number (in decimal representation) with n prime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 16, 17, 115, 121, 509, 821, 3251, 4721, 5749, 22760, 25301, 41673, 142950, 173819, 291714, 920561, 1222716, 2041709, 4450031, 8559017, 9350687, 14295199, 31150219, 50568439, 71502954, 100066398, 218051538, 353979075, 500526787, 702815371, 1512442643

Offset: 0

Hieronymus Fischer, Nov 22 2012

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=7*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 7^j = (7^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-7) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-7 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 7.

			a(1) = 2 = 2_7, since 2 is the least number with 1 prime substring in base-7 representation.
a(2) = 16 = 22_7, since 16 is the least number with 2 prime substrings in base-7 representation (2 times 2_7=2).
a(3) = 17 = 23_7, since 17 is the least number with 3 prime substrings in base-7 representation (2_7, 3_7, and 23_7).
a(5) = 121 = 232_7, since 121 is the least number with 5 prime substrings in base-7 representation (2 times 2_7, 3_7, 23_7=17, and 32_7=23).
a(6) = 509 = 1325_7, since 509 is the least number with 6 prime substrings in base-7 representation (2_7, 3_7, 5_7, 25_7=19, 32_7=23, and 1325_7=509).

Hieronymus Fischer, Table of n, a(n) for n = 0..40

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217302-A217309.

a(n) > 7^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= (7^n - 1)/3, n>0.
a(n+1) <= 7*a(n) + 2.

A217114 Greatest number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

11, 59, 239, 251, 751, 1007, 1019, 3823, 4079, 4055, 16111, 16087, 16319, 16367, 48991, 64351, 65263, 65269, 65471, 253919, 260959, 261079, 261847, 261871, 916319, 1043839, 1047391, 1044463, 1047511, 3665279, 3140991, 4189567, 4118519, 4177759, 4189565, 4193239, 14661117

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 4^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-4 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. This proves the statement of existence. Proof of finiteness: Each 3-digit base-4 number has at least 1 nonprime substring. Hence, each 3(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 4^(3n+2) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 11, since 11 = 23_4 (base-4) is the greatest number with zero nonprime substrings in base-4 representation.
a(1) = 59 = 323_4 has 6 substrings in base-4 representation (2, 3, 3, 23, 32 and 323), only 32_4=14 is a nonprime substring. 59 is the greatest such number with 1 nonprime substring.
a(2) = 239 = 3233_4 has 10 substrings in base-4 representation (2, 3, 3, 23, 32, 323, 233 and 3233), exactly 2 of them are nonprime substrings (32_4=14 and 33_4=15), and there is no greater number with 2 nonprime substrings in base-4 representation.
a(11) = 16087 = 3323113_4 has 28 substrings in base-4 representation. The base-4 nonprime substrings are 1, 1, 32, 33, 231, 332, 3113, 3231, 32311, 33321 and 323113. There is no greater number with 11 nonprime substrings in base-4 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..100

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217104(n).
a(n) >= A217304(A000217(A110591(a(n)))-n).
a(n) <= 4^(n+2).
a(n) <= 4^min((n + 6)/2, 9*floor((n+18)/19)).
a(n) <= 64*4^(n/2).
a(n+m+1) >= 4*a(n), where m := floor(log_4(a(n))) + 1.

cp's OEIS Frontend

A217106 Minimal number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

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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).

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A217108 Minimal number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

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A217113 Greatest number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

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A217118 Greatest number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

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A217304 Minimal natural number (in decimal representation) with n prime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

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A217305 Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

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A217306 Minimal natural number (in decimal representation) with n prime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

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A217307 Minimal natural number (in decimal representation) with n prime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

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