A039996 -id:A039996 - cp's OEIS frontend

A213311 Numbers with exactly 4 nonprime substrings (substrings with leading zeros are considered to be nonprime).

103, 107, 111, 112, 115, 119, 122, 125, 129, 130, 134, 136, 138, 143, 147, 151, 152, 155, 159, 163, 170, 174, 176, 178, 183, 191, 192, 195, 199, 202, 203, 205, 207, 212, 215, 219, 220, 221, 224, 226, 228, 242, 245, 250

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 6-digit number has at least 4 nonprime substrings, and each 4-digit number has at least 1 nonprime substring. Thus, each 10-digit number has at least 5 nonprime substrings. Consequently, there is a boundary b, such that all numbers >= b have more than 4 nonprime substrings.

The first term is a(1)=103=A213302(4). The last term is a(653)=373379=A213300(4).

			a(1) = 103, since 103 has 4 nonprime substrings (0, 03, 1, 10).
a(653) = 373379, since there are 4 nonprime substrings (9, 33, 3379, 7337).

Hieronymus Fischer, Table of n, a(n) for n = 1..653

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

Cf. A035244, A079307, A213300 - A213321.

A213312 Numbers with exactly 5 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

101, 102, 105, 109, 110, 114, 116, 118, 120, 121, 124, 126, 128, 141, 142, 145, 149, 150, 154, 156, 158, 161, 162, 165, 181, 182, 185, 187, 189, 190, 194, 196, 198, 200, 201, 204, 206, 208, 209, 210, 214, 216, 218, 240

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 7-digit number has at least 6 nonprime substrings. Thus, each number with more than 7 digits has >= 6 nonprime substrings, too. Consequently, there is a boundary b<10^6, such that all numbers > b have more than 5 nonprime substrings.

The first term is a(1)=101=A213302(5). The last term is a(1330)=831373=A213300(5).

			a(1)=101, since 101 has 5 nonprime substrings (0, 01, 1, 1, 10).
a(1330)= 831373, since there are 5 nonprime substrings (1, 8, 831, 8313, 31373).

Hieronymus Fischer, Table of n, a(n) for n = 1..1330

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

Cf. A035244, A079307, A213300 - A213321.

A213313 Numbers with exactly 6 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

100, 104, 106, 108, 140, 144, 146, 148, 160, 164, 166, 168, 169, 180, 184, 186, 188, 400, 404, 406, 408, 440, 444, 446, 448, 460, 464, 466, 468, 469, 480, 481, 484, 486, 488, 490, 494, 496, 498, 600, 604, 606, 608, 609

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 8-digit number has at least 10 nonprime substrings. Thus, each number with more than 8 digits has >= 10 nonprime substrings, too. Consequently, there is a boundary b<10^7, such that all numbers > b have more than 6 nonprime substrings.

The first term is a(1)=100=A213302(6). The last term is a(2351)=3733797=A213300(6).

			a(1)=100, since 100 has 6 nonprime substrings (0, 0, 00, 1, 10, 100).
a(2351)= 3733797, since there are 6 nonprime substrings (9, 33, 3379, 7337, 733797, 3733797).

Hieronymus Fischer, Table of n, a(n) for n = 1..2351

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

Cf. A035244, A079307, A213300 - A213321.

Select[Range[700],Count[FromDigits/@Flatten[Table[Partition[ IntegerDigits[ #],n,1], {n, IntegerLength[#]}],1],?(!PrimeQ[#]&)]==6&] (* _Harvey P. Dale, Apr 08 2019 *)

A213314 Numbers with exactly 7 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1017, 1019, 1023, 1032, 1035, 1039, 1053, 1071, 1072, 1075, 1077, 1079, 1093, 1107, 1109, 1111, 1112, 1115, 1119, 1122, 1125, 1143, 1147, 1152, 1155, 1159, 1170, 1174, 1176, 1178, 1181, 1183, 1187, 1191, 1192, 1195

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 8-digit number has at least 10 nonprime substrings. Thus, each number with more than 8 digits has >= 10 nonprime substrings, too. Consequently, there is a boundary b<10^7, such that all numbers > b have more than 7 nonprime substrings.

The first term is a(1)=1017=A213302(7). The last term is a(4362)=3733739=A213300(7).

			a(1)=1017, since 1017 has 7 nonprime substrings (0, 1, 1, 01, 10, 017, 1017).
a(4362)= 3733739 since there are 7 nonprime substrings (9, 33, 39, 7337, 73373, 373373, 733739).

Hieronymus Fischer, Table of n, a(n) for n = 1..4362

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

Cf. A035244, A079307, A213300 - A213321.

A213315 Numbers with exactly 8 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1011, 1012, 1015, 1021, 1022, 1025, 1027, 1029, 1030, 1034, 1036, 1038, 1043, 1047, 1051, 1052, 1055, 1057, 1059, 1061, 1063, 1067, 1070, 1074, 1076, 1078, 1083, 1087, 1091, 1092, 1095, 1101, 1102, 1105, 1110, 1114

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 8-digit number has at least 10 nonprime substrings. Thus, each number with more than 8 digits has >= 10 nonprime substrings, too. Consequently, there is a boundary b<10^7, such that all numbers > b have more than 8 nonprime substrings.

The first term is a(1)=1011=A213302(8). The last term is a(7483)=8313733=A213300(8).

			a(1)=1011, since 1011 has 8 nonprime substrings (0, 1, 1, 1, 01, 10, 011, 1011).
a(7483)= 8313733 since there are 8 nonprime substrings (1, 8, 33, 831, 8313, 13733, 31373, 313733).

Hieronymus Fischer, Table of n, a(n) for n = 1..7483

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

Cf. A035244, A079307, A213300 - A213321.

A213316 Numbers with exactly 9 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1002, 1003, 1005, 1007, 1009, 1010, 1014, 1016, 1018, 1020, 1024, 1026, 1028, 1041, 1042, 1045, 1049, 1050, 1054, 1056, 1058, 1062, 1065, 1069, 1082, 1085, 1089, 1090, 1094, 1096, 1098, 1099, 1100, 1104, 1106, 1108, 1140, 1144, 1146, 1148

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 8-digit number has at least 10 nonprime substrings. Thus each number with more than 8 digits has >= 10 nonprime substrings, too. Consequently there is a boundary b<10^7 such that all numbers > b have more than 9 nonprime substrings.

The first term is a(1) = 1002 = A213302(9). The last term is a(12411) = 9973331 = A213300(9).

			a(1) = 1002 is in the sequence, since 1002 has 9 nonprime substrings (0,  0, 1, 00, 02, 10, 002, 100, 1002).
a(12411) = 9973331 is in the sequence since there are 9 nonprime substrings (1, 9, 9, 33, 33, 99, 333, 973, 97333).

Hieronymus Fischer, Table of n, a(n) for n = 1..12411

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

Cf. A035244, A079307, A213300 - A213321.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A213311 Numbers with exactly 4 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213312 Numbers with exactly 5 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213313 Numbers with exactly 6 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213314 Numbers with exactly 7 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213315 Numbers with exactly 8 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213316 Numbers with exactly 9 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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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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