A213300 -id:A213300 - cp's OEIS frontend

A213317 Numbers with exactly 10 nonprime substrings (substrings with leading zeros are considered to be nonprime).

1000, 1001, 1004, 1006, 1008, 1040, 1044, 1046, 1048, 1060, 1064, 1066, 1068, 1080, 1081, 1084, 1086, 1088, 1400, 1404, 1406, 1408, 1440, 1444, 1446, 1448, 1460, 1464, 1466, 1468, 1469, 1480, 1484, 1486, 1488, 1600

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 9-digit number has at least 15 nonprime substrings. Thus, each number with more than 9 digits has >= 15 nonprime substrings, too. Consequently, there is a boundary b<10^9, such that all numbers > b have more than 10 nonprime substrings.

The first term is a(1)=1000=A213302(10). The last term is a(20230)=37337397=A213300(10).

			a(1)=1000, since 1000 has 10 nonprime substrings (0, 0, 0, 1, 00, 00, 10, 000, 100, 1000).
a(20230)= 37337397, since there are 10 nonprime substrings (9, 33, 39, 7337, 7397, 73373, 373373, 733739, 7337397, 37337397).

Hieronymus Fischer, Table of n, a(n) for n = 1..20230

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

A213318 Numbers with exactly 11 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

10037, 10103, 10111, 10117, 10123, 10127, 10130, 10134, 10136, 10138, 10151, 10153, 10157, 10159, 10163, 10167, 10171, 10172, 10175, 10191, 10192, 10195, 10199, 10213, 10217, 10227, 10229, 10231, 10232, 10235, 10239, 10243

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 9-digit number has at least 15 nonprime substrings. Thus, each number with more than 9 digits has >= 15 nonprime substrings, too. Consequently, there is a boundary b<10^9, such that all numbers > b have more than 11 nonprime substrings.

The first term is a(1)=10037=A213302(11). The last term is a(32869)=82337397=A213300(11).

			a(1)= 10037, since 10037 has 11 nonprime substrings (0, 0, 1, 00, 03, 10, 003, 037, 100, 0037, 1003).
a(32869)= 82337397, since there are 11 nonprime substrings (8, 9, 33, 39, 82, 2337, 7397, 23373, 82337, 233739, 82337397).

Hieronymus Fischer, Table of n, a(n) for n = 1..32869

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079307, A213300 - A213321.

A213319 Numbers with exactly 12 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

10023, 10053, 10067, 10073, 10079, 10093, 10097, 10107, 10112, 10115, 10119, 10122, 10125, 10129, 10141, 10143, 10147, 10152, 10155, 10170, 10174, 10176, 10178, 10181, 10183, 10190, 10194, 10196, 10198, 10212, 10215, 10219

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 9-digit number has at least 15 nonprime substrings. Thus, each number with more than 9 digits has >= 15 nonprime substrings, too. Consequently, there is a boundary b<10^9, such that all numbers > b have more than 12 nonprime substrings.

The first term is a(1)=10023=A213302(12). The last term is a(51477)=99733313=A213300(12).

			a(1)=10023, since 10023 has 12 nonprime substrings (0, 0, 1, 00, 02, 10, 002, 023, 100, 0023, 1002, 10023).
a(51477)=99733313, since there are 11 nonprime substrings (1, 9, 9, 33, 33, 99, 333, 973, 33313, 97333, 733313, 99733313).

Hieronymus Fischer, Table of n, a(n) for n = 1..51477

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

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.

A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

67, 88, 442, 567, 2213, 2837, 3067, 11068, 14713, 15337, 15338, 57943, 73568, 77213, 76697, 289717, 280338, 370443, 386068, 386587, 389713, 1852217, 1524067, 1898442, 1930342, 1932943, 1948568, 7242943, 9261088, 9664717, 9586567, 9654712, 9710942, 9742849, 46305443

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} 5^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-5 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-5 number has at least 2 nonprime substrings. Hence, each m := 4*floor((n+2)/2)-digit number has at least 2*(m/4) = 2*floor((n+2)/2) >= n+1 nonprime substrings. Consequently, there is a boundary b < 5^(m-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) = 67, since 67 = 232_5 (base-5) is the greatest number with zero nonprime substrings in base-5 representation.
a(1) = 88 = 323_5 has 6 substrings in base-5 representation (2, 2, 3, 23, 32, 323), the only nonprime substring is 323_5. 88 is the greatest number with 1 nonprime substring.
a(2) = 442 = 3232_5 has 10 substrings in base-5 representation (2, 2, 3, 3, 23, 32, 32, 232, 323 and 3232), exactly 2 of them are nonprime substrings (323_5=88 and 3232_5=442), and there is no greater number with 2 nonprime substrings in base-5 representation.
a(5) = 2837 = 42322_5 has 5 nonprime substrings in base-5 representation, these are 4, 22, 42, 322 and 4232, all the other substrings are prime. There is no greater number with 5 nonprime substrings in base-5 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..70

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) >= A217105(n).
a(n) >= A217305(A000217(A110592(a(n)))-n).
a(n) <= 5^(n+3).
a(n) <= 5^(4*floor(n/2)), n>1.
a(n) <= 5^min((n + 6)/2, 9*floor((n+20)/21)).
a(n) <= 125*5^(n/2).
With m := floor(log_5(a(n))) + 1:
a(n+m+1) >= 5*a(n), if a(n)!=1 (mod 5).
a(n+m) >= 5*a(n), if a(n)=1 (mod 5).

A217116 Greatest number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

23, 839, 5039, 6983, 7127, 42743, 45863, 46199, 275183, 279143, 277199, 1088531, 1674863, 1651103, 1674859, 6713711, 9906599, 10045559, 10072943, 39190247, 40278647, 60273359, 60295079, 60294239, 60437659, 241671887, 342609527, 359245007, 361640159, 362625959

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} 6^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-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. This proves the statement of existence. Proof of finiteness: Each 3-digit base-6 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 < 6^(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.

			(0) = 23, since 23 = 35_6 (base-6) is the greatest number with zero nonprime substrings in base-6 representation.
a(1) = 839 = 3515_6 has 1 nonprime substring in base-6 representation (= 1). All the other base-6 substrings (3, 5, 15_6=11, 35_6=23, 51_6=31, 351_6=139, 515_6=191 and 3515_6=839) are prime substrings. 839 is the greatest number with 1 nonprime substring.
a(2) = 5039 = 35155_6 has 2 nonprime substrings in base-6 representation (1 and 55_6=35), and there is no greater number with 2 nonprime substrings in base-6 representation.
a(5) = 42743 = 525515_6 has 5 nonprime substrings in base-6 representation, these are 1, 52_6=32, 55_6=35, 5515_6=1271 and 52551_6=7123, and there is no greater number with 5 nonprime substrings in base-6 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..55

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) >= A217106(n).
a(n) >= A217306(A000217(num_digits_6(a(n)))-n), where num_digits_6(x) is the number of digits of the base-6 representation of x.
a(n) <= 6^min(n+3, 7*floor((n+7)/8)).
a(n) <= 216*6^n.
a(n+m+1) >= 6*a(n), where m := floor(log_6(a(n))) + 1.

A217117 Greatest 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

37, 331, 317, 2322, 2389, 15259, 16260, 16728, 100291, 113825, 116101, 117109, 796777, 796781, 819719, 823003, 4753901, 5577444, 5738035, 5738039, 5761027, 31150219, 39041113, 39336580, 40166250, 40326841, 40336249, 218051538, 273271861

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} 7^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-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. This proves the statement of existence. Proof of finiteness: Each 3-digit base-7 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 < 7^(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) = 37, since 37 = 52_7 (base-7) is the greatest number with zero nonprime substrings in base-7 representation.
a(1) = 331 = 652_7 has 1 nonprime substring in base-7 representation (= 6). All the other base-7 substrings (2, 5, 52_7=37, 65_7=47 and 652_7=331) are prime substrings. 331 is the greatest number with 1 nonprime substring.
a(2) = 317 = 632_7 has 6 substrings in base-7 representation (2, 3, 6, 32, 63 and 632), exactly 2 of them are nonprime substrings (6 and 32_6=20), and there is no greater number with 2 nonprime substrings in base-7 representation.
a(8) = 100291 = 565252_3 has 8 nonprime substrings in base-7 representation, these are 6, 252_7, 525_7, 565_7, 5252_7, 5652_7, 6525_7 and 65252_7. There is no greater number with 8 nonprime substrings in base-7 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) >= A217107(n).
a(n) >= A217307(A000217(num_digits_7(a(n)))-n), where num_digits_7(x) is the number of digits of the base-7 representation of x.
a(n) <= 7^min(n+2, 5*floor((n+4)/5)).
a(n) <= 7^(n+2).
a(n) <= 7^min(3 + n/2, 8*floor((n+15)/16)).
a(n) <= 343*7^(n/2).
With m := floor(log_7(a(n))) + 1:
a(n+m+1) >= 7*a(n), if a(n)!=1 (mod 7).
a(n+m) >= 7*a(n), if a(n)=1 (mod 7).

A330007 Base-24 integers whose substrings are primes (written in base 10).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 53, 59, 61, 67, 71, 79, 83, 89, 127, 131, 137, 139, 173, 179, 181, 191, 269, 271, 277, 281, 283, 317, 331, 419, 421, 431, 461, 463, 467, 479, 557, 563, 569, 571, 1279, 1283, 1289, 1291, 1423, 1429, 1433, 1483, 1613, 1619, 1709, 1721, 1723, 1901, 1907, 1997, 1999, 2011, 3061, 3163, 3299

Offset: 1

Onno M. Cain, Nov 26 2019

The largest such number is a(103)=266003, which is written (19|5|19|11)_24 in base 24.

We might call these numbers "substrimes" (= substring-primes) since the term (1) is concise, (2) is pronounceable, and (3) keeps these numbers distinct in communication from different but similar sequences (see Crossrefs).

			a(64) = 3299 = (5|17|11)_24 is in the sequence because 5, 17, 11, (5|17)_24=137, (17|11)=419, and 3299 itself are all prime.

Onno M. Cain, Python Code

Cf. A024773, A085823, A085822, A213300.

With[{b = 24}, Select[Range[b^4], Function[{s, n}, AllTrue[Flatten@ Array[FromDigits[#, b] & /@ Partition[s, #, 1] &, n], PrimeQ]] @@ {#, Length@ #} &@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 15 2019 *)

Python
```
# see Cain link
```

cp's OEIS Frontend

A213317 Numbers with exactly 10 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213318 Numbers with exactly 11 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213319 Numbers with exactly 12 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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

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A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

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A217116 Greatest 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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A217117 Greatest 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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A330007 Base-24 integers whose substrings are primes (written in base 10).

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