A079307 -id:A079307 - cp's OEIS frontend

A213320 Numbers such that the number of nonprime substrings equals the number of digits (substrings with leading zeros are considered to be nonprime).

1, 4, 6, 8, 9, 11, 12, 15, 19, 20, 21, 24, 26, 28, 30, 34, 36, 38, 39, 41, 42, 45, 50, 51, 54, 56, 58, 61, 62, 63, 65, 70, 74, 76, 78, 82, 85, 87, 89, 92, 93, 95, 117, 123, 127, 132, 133, 135, 139, 153, 157, 167, 171, 172, 175

Offset: 1

Hieronymus Fischer, Aug 26 2012

Also numbers such that the number of prime substrings is A000217(m-1) = m(m-1)/2, where m is the number of digits.
The sequence is finite. Proof: Let p be a number >= 10^17 and let m = 9k+j be the number of digits of p, where k = floor(m/9) >= 2 and j = m mod 9. Since each 9-digit number has at least 15 nonprime substrings, it follows that p has at least 15k = 9k + 6k > 9k + j = m nonprime substrings (since 6k >= 12> j for k >= 2). Consequently, no number >= 10^17 can be a term of the sequence.
The last term is a(858)=3733739. Proof: Each 9-digit number has at least 15 nonprime substrings, thus, the numbers 10^8 <= p < 10^14 also have at least 15 nonprime substrings and therefore cannot be terms of the sequence. Same is true for numbers 10^14 <= p < 10^17 since each 6-digit number has at least 4 nonprime substrings, and thus each number with >= 15 digits has at least 15+4 = 19 nonprime substrings. Since each 8-digit number has at least 10 nonprime substrings, it follows that the last term of the sequence must be less than 10^7. By direct search we find a(858) = 3733739.

			a(1) = 1, since 1 has 1 nonprime substrings.
a(43) = 117, since 117 has 3 digits and also 3 nonprime substrings (1, 1, 117).

Hieronymus Fischer, Table of n, a(n) for n = 1..858 (full sequence)

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

Cf. A035244, A079307, A213300 - A213321.

Typo in example corrected, Hieronymus Fischer, Sep 11 2012

A213308 Numbers with exactly one nonprime substring (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 4, 6, 8, 9, 13, 17, 22, 25, 27, 29, 31, 32, 33, 35, 43, 47, 52, 55, 57, 59, 67, 71, 72, 75, 77, 79, 83, 97, 137, 173, 223, 233, 237, 313, 317, 337, 353, 379, 523, 537, 673, 733, 737, 773, 797, 1373, 3137, 3373, 3733, 3797

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 5-digit number has at least 2 nonprime substrings. Thus, each number with more than 5 digits has >= 2 nonprime substrings, too. Consequently, there is a boundary b<10^4, such that all numbers > b have at least 2 nonprime substrings.

The first term is a(1)=1=A213302(1). The last term is a(51)=3797=A213300(1).

			a(1)=1, since 1 has one nonprime substring.
a(51)=3797, since the only nonprime substring of 3797 is 9.

Hieronymus Fischer, Table of n, a(n) for n = 1..51

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

Cf. A035244, A079307, A213300 - A213321.

A213309 Numbers with exactly 2 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

11, 12, 15, 19, 20, 21, 24, 26, 28, 30, 34, 36, 38, 39, 41, 42, 45, 50, 51, 54, 56, 58, 61, 62, 63, 65, 70, 74, 76, 78, 82, 85, 87, 89, 92, 93, 95, 113, 131, 179, 197, 227, 229, 231, 232, 235, 239, 253, 257, 271, 273, 277, 283

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 6-digit number has at least 4 nonprime substrings. Thus, each number with more than 6 digits has >= 4 nonprime substrings, too. Consequently, there is a boundary b<10^5, such that all numbers > b have more than 2 nonprime substrings.

The first term is a(1)=11=A213302(2). The last term is a(130)=37337=A213300(2).

			a(1)=11, since 11 has 2 nonprime substrings.
a(130)= 37337, since there are 2 nonprime substrings (33 and 337).

Hieronymus Fischer, Table of n, a(n) for n = 1..130

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

Cf. A035244, A079307, A213300 - A213321.

A213310 Numbers with exactly 3 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

10, 14, 16, 18, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 81, 84, 86, 88, 90, 91, 94, 96, 98, 99, 117, 123, 127, 132, 133, 135, 139, 153, 157, 167, 171, 172, 175, 177, 193, 211, 213, 217, 222, 225, 230, 234, 236, 238, 241

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 6-digit number has at least 4 nonprime substrings. Thus, each number with more than 6 digits has >= 4 nonprime substrings, too. Consequently, there is a boundary b<10^5, such that all numbers > b have more than 3 nonprime substrings.

The first term is a(1)=10=A213302(3). The last term is a(310)=73373=A213300(3).

			a(1)=10, since 10 has 3 nonprime substrings (0, 1, 10).
a(310)= 73373, since there are 3 nonprime substrings (33, 7337 and 73373).

Hieronymus Fischer, Table of n, a(n) for n = 1..310

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

Cf. A035244, A079307, A213300 - A213321.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A213320 Numbers such that the number of nonprime substrings equals the number of digits (substrings with leading zeros are considered to be nonprime).

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A213308 Numbers with exactly one nonprime substring (substrings with leading zeros are considered to be nonprime).

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A213309 Numbers with exactly 2 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213310 Numbers with exactly 3 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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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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