A213302 -id:A213302 - cp's OEIS frontend

A213300 Largest number with n nonprime substrings (substrings with leading zeros are considered to be nonprime).

373, 3797, 37337, 73373, 373379, 831373, 3733797, 3733739, 8313733, 9973331, 37337397, 82337397, 99733313, 99733317, 99793373, 733133733, 831373379, 997333137, 997337397, 997933739, 7331337337, 8313733797, 9733733797, 9973331373, 9979337397, 9982337397

Offset: 0

Hieronymus Fischer, Aug 26 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is nonempty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 10^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n - k(k+1)/2. For n=0,1,2,3,... the m(n) 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 existence. Proof of finiteness: Each 4-digit 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 < 10^(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.
The following statements hold true:
For all n>=0 there are minimal numbers with n nonprime substrings (cf. A213302 - A213304).
For all n>=0 there are maximal numbers with n nonprime substrings (= A213300 = this sequence).
For all n>=0 there are minimal numbers with n prime substrings (cf. A035244).
The greatest number with n prime substrings does not exist. Proof: If p is a number with n prime substrings, than 10*p is a greater number with n prime substrings.
Comment from N. J. A. Sloane, Sep 01 2012: it is a surprise that any number greater than 373 has a nonprime substring!

			a(0)=373, since 373 is the greatest number such that all substrings are primes, hence it is the maximal number with 0 nonprime substrings.
a(1)=3797, since the only nonprime substring of 3797 is 9 and all greater numbers have more than 1 nonprime substrings.
a(2)=37337, since the nonprime substrings of 37337 are 33 and 7337 and all greater numbers have > 2 nonprime substrings.

Hieronymus Fischer, Table of n, a(n) for n = 0..32

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

Cf. A035244, A079307, A213301 - A213321.

a(n) >= A035244(A000217(A055642(a(n)))-n).

A217102 Minimal number (in decimal representation) with n nonprime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 7, 5, 4, 11, 10, 12, 8, 22, 21, 19, 17, 16, 60, 39, 37, 34, 36, 32, 83, 71, 74, 69, 67, 66, 64, 143, 139, 141, 135, 134, 131, 130, 128, 283, 271, 269, 263, 267, 262, 261, 257, 256, 541, 539, 527, 526, 523, 533, 519, 514, 516, 512, 1055, 1053, 1047, 1067

Offset: 1

Hieronymus Fischer, Dec 12 2012

There are no numbers with zero nonprime substrings in binary representation. For all bases > 2 there is always a number (=2) with zero nonprime substrings (Cf. A217103-A217109, A213302).

If p is a number with k prime substrings and d digits (in binary representation), p even, m>=d, than b := p*2^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(1) = 1, since 1 = 1_2 is the least number with 1 nonprime substring in binary representation.
a(2) = 2, since 2 = 10_2 is the least number with 2 nonprime substrings in binary representation (0 and 1).
a(3) = 7, since 7 = 111_2 is the least number with 3 nonprime substrings in binary representation (3-times 1, the prime substrings are 2-times 11 and 111).
a(10) = 22, since 22 = 10110_2 is the least number with 10 nonprime substrings in binary representation, these are 0, 0, 1, 1, 1, 01, 011, 110, 0110 and 10110 (remember, that substrings with leading zeros are considered to be nonprime).

Hieronymus Fischer, Table of n, a(n) for n = 1..2015

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217103-A217109, A217302-A217309.

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

A213305 Minimal prime with n nonprime substrings (Version 1: substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 13, 11, 127, 103, 101, 1013, 1019, 1021, 1009, 10177, 10037, 10067, 10007, 10009, 100237, 100271, 100153, 100043, 100003, 100049, 1001173, 1000313, 1000037, 1000033, 1000039, 1000003, 1000081, 10000379, 10001237, 10000223

Offset: 0

Hieronymus Fischer, Aug 26 2012

			a(0) = 2, since 2 is the least prime with zero nonprime substrings.
a(1) = 13, since 13 is the least prime with exactly 1 (“1”) nonprime substrings.
a(2) = 11, since 11 is the least prime with exactly 2 (“1” and “1”) nonprime substrings.

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

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

Cf. A035244, A079307, A213300 - A213321.

a(n) > 10^floor((sqrt(8*n+1)-1)/2) for n>0.
a(m(m+1)/2) > 10^m, m>0.
a(n) >= A213302(n).

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.

cp's OEIS Frontend

A213300 Largest number with n nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A217102 Minimal number (in decimal representation) with n nonprime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

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A213305 Minimal prime with n nonprime substrings (Version 1: 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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