A085823 -id:A085823 - cp's OEIS frontend

2, 1, 11, 10, 103, 101, 100, 1017, 1011, 1002, 1000, 10037, 10023, 10007, 10002, 10000, 100137, 100073, 100023, 100003, 100002, 100000, 1000313, 1000037, 1000033, 1000023, 1000003, 1000002, 1000000, 10000337, 10000223, 10000137, 10000037, 10000023, 10000013, 10000002, 10000000, 100001733

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 not empty. Proof: Define m(n)=2*sum_{j=i..k} 10^j, where k=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). 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, which proves the statement.

The 3 versions according to A213302 - A213304 are quite different. Example: 1002 has 9 nonprime substrings in version 1 (0, 0, 00, 02, 002, 1, 10 100, 1002), in version 2 there are 6 nonprime substrings (02, 002, 1, 10, 100, 1002) and there are 4 nonprime substrings in version 3 (1, 10, 100, 1002).

			a(0)=2, since 2 is the least number with zero nonprime substrings.
a(1)=1, since 1 has 1 nonprime substrings.
a(2)=11, since 11 is the least number with 2 nonprime substrings.
a(3)=10, since 10 is the least number with 3 nonprime substrings, these are 1, 0 and 10 (‘0’ will be counted).

Hieronymus Fischer, Table of n, a(n) for n = 0..200

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

Cf. A035244, A079307, A213300 - A213321.

a(n) >= 10^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number > 0 (cf. A000217).

a(A000217(n)) = 10^(n-1), n>0.

a(A000217(n)-k) >= 10^(n-1)+k, n>0, 0<=k

a(A000217(n)-1) = 10^(n-1)+2, n>3, provided 10^(n-1)+1 is not a prime (which is proved to be true for all n-1 <= 50000 (cf. A185121) except n-1=16384 and is generally true for n-1 unequal to a power of 2).

a(A000217(n)-k) = 10^(n-1)+p, where p is the minimal number such that 10^(n-1) + p, has k prime substrings, n>0, 0<=k

Min(a(A000217(n)-k-i), 0<=i<=m) <= 10^(n-1)+p, where p is the minimal number with k prime substrings and m is the number of digits of p, and k+m

Min(a(A000217(n)-k-i), 0<=i<=A055642(A035244(k)) <= 10^(n-1)+A035244(k).

a(A000217(n)-k) <= 10^(n-1)+max(p(i), k<=i<=k+m), where p(i) is the minimal number with i prime substrings and m is the number of digits of p(i), and k+m

a(A000217(n)-k) <= 10^(n-1)+max(A035244(i), k<=i<=k+ A055642(i).

a(n) <= A213305(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.

A217109 Minimal number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 12, 9, 83, 84, 81, 748, 740, 731, 729, 6653, 6581, 6563, 6564, 6561, 59222, 59069, 59068, 59051, 59052, 59049, 531614, 531569, 531464, 531460, 531452, 531443, 531441, 4784122, 4783142, 4783147, 4783070, 4782989, 4782971, 4782972, 4782969, 43048283

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} 9^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-9 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-9 representation), m>=d, than b := p*9^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_9 is the least number with zero nonprime substrings in base-9 representation.
a(1) = 1, since 1 = 1_9 is the least number with 1 nonprime substring in base-9 representation.
a(2) = 12, since 12 = 13_9 is the least number with 2 nonprime substrings in base-9 representation (1 and 13).
a(3) = 9, since 9 = 10_9 is the least number with 3 nonprime substrings in base-9 representation (0, 1 and 10).
a(4) = 83, since 83 = 102_9 is the least number with 4 nonprime substrings in base-9 representation, these are 0, 1, 10, and 02 (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) >= 9^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 9^(n-1), n>0.
a(A000217(n)-k) >= 9^(n-1) + k, 0<=k0.
a(A000217(n)-k) = 9^(n-1) + p, where p is the minimal number >= 0 such that 9^(n-1) + p, has k prime substrings in base-9 representation, 0<=k0.

A202262 Composite numbers in which all substrings are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 444, 446, 448, 464, 466, 468, 469, 484, 486, 488, 494, 496, 498, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 694, 696, 698, 699, 844, 846, 848, 849, 864, 866, 868, 869, 884, 886

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A062115, A202260, A029581.
Supersequence of A202265.
This is a 10-automatic sequence, see A071062. - Charles R Greathouse IV, Jan 01 2012

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A202262
Index entries for 10-automatic sequences

Cf. A085823 (primes in which all substrings are primes), A068669 (noncomposite numbers in which all substrings are noncomposite), A062115 (nonprimes in which all substrings are nonprimes).

Cf. A010051.

sub[n_] := Block[{d = IntegerDigits[n]}, Union@ Reap[ Do[Sow@ FromDigits@ Take[d, {i, j}], {j, Length@ d}, {i, j}]][[2, 1]]]; Select[ Range@ 900, Union[{4, 6, 8, 9}, IntegerDigits[#]] == {4, 6, 8, 9} && AllTrue[sub[#], CompositeQ] &] (* Giovanni Resta, Dec 20 2019 *)

PARI
```
See Links section.
```

Data corrected by Reinhard Zumkeller, May 05 2012
Data corrected by Rémy Sigrist, Dec 19 2019
Incorrect Haskell program deleted by M. F. Hasler, Dec 20 2019

A217112 Greatest 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, 3, 7, 6, 15, 14, 31, 29, 30, 63, 61, 62, 127, 54, 125, 126, 255, 117, 251, 254, 189, 511, 479, 509, 510, 379, 502, 1023, 1021, 1007, 1022, 958, 1018, 1014, 2047, 2045, 1791, 2046, 2042, 2027, 2037, 4091, 4095, 4063, 3069, 4094, 4090, 4085, 8159, 8187, 8191, 8189, 8127

Offset: 1

Hieronymus Fischer, Dec 20 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.

The set of numbers with n nonprime substrings is finite. Proof: Evidently, each 1-digit binary number represents 1 nonprime substring. Hence, each (n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 2^n, such that all numbers > b have more than n nonprime substrings.

			(1) = 1, since 1 = 1_2 (binary) is the greatest number with 1 nonprime substring.
a(2) = 3 = 11_2 has 3 substrings in binary representation (1, 1 and 11), two of them are nonprime substrings (1 and 1), and 11_2 = 3 is the only prime substrings. 3 is the greatest number with 2 nonprime substrings.
a(8) = 29 = 11101_2 has 15 substrings in binary representation (0, 1, 1, 1, 1, 11, 11, 10, 01, 111, 110, 101, 1110, 1101, 11101), exactly 8 of them are nonprime substrings (0, 1, 1, 1, 1, 01, 110, 1110). There is no greater number with 8 nonprime substrings in binary representation.
a(14) = 54 = 110110_2 has 21 substrings in binary representation, only 7 of them are prime substrings (10, 10, 11, 11, 101, 1011, 1101), which implies that exactly 14 substrings must be nonprime. There is no greater number with 14 nonprime substrings in binary representation.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217113 - A217119.
Cf. A217302 - A217309.

a(n) >= A217102(n).
a(n) >= A217302(A000217(A070939(a(n)))-n).
Example: a(9)=30=11110_2, A000217(A070939(31))=15, hence, a(9)>=A217302(15-9)=27.
a(n) <= 2^n.
a(n) <= 2^min(6 + n/6, 20*floor((n+125)/126)).
a(n) <= 64*2^(n/6).
With m := floor(log_2(a(n))) + 1:
a(n+m+1) >= 2*a(n), if a(n) is even.
a(n+m) >= 2*a(n), if a(n) is odd.

A217119 Greatest number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

47, 428, 1721, 6473, 14033, 35201, 58961, 58967, 465743, 530701, 530710, 1733741, 4250788, 4723108, 4776398, 25051529, 37327196, 42450640, 42986860, 42987589, 42996409, 225463817, 382055767, 382571822, 386888308, 386888419, 387356789

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} 9^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-9 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-9 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 < 9^(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) = 47, since 47 = 52_9 (base-9) is the greatest number with zero nonprime substrings in base-9 representation.
a(1) = 428 = 525_9 has 1 nonprime substring in base-9 representation (= 525_9). All the other base-9 substrings (2, 5, 5, 25, 52) are prime substrings. 525_9 is the greatest number with 1 nonprime substring.
a(2) = 1721 = 2322_9 has 10 substrings in base-9 representation, exactly 2 of them are nonprime substrings (22_9 and 23_3=8), and there is no greater number with 2 nonprime substrings in base-9 representation.
a(7) = 58967= 88788_9 has 15 substrings in base-9 representation, exactly 7 of them are nonprime substrings (4-times 8, 2-times 88, and 8788), and there is no greater number with 7 nonprime substrings in base-9 representation.

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. A217102 - A217109, A217112 - A217118.
Cf. A217302 - A217309.

a(n) >= A217109(n).
a(n) >= A217309(A000217(num_digits_9(a(n)))-n), where num_digits_9(x)=floor(log_9(x))+1 is the number of digits of the base-9 representation of x.
a(n) <= 9^(n+2).
a(n) <= 9^min(n+2, 6*floor((n+7)/8)).
a(n) <= 9^((3/4)*(n + 3)).
a(n+m+1) >= 9*a(n), where m := floor(log_9(a(n))) + 1.

A166504 Slime numbers: numbers which are the concatenation of primes, with "leading zeros" allowed.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 47, 52, 53, 55, 57, 59, 61, 67, 71, 72, 73, 75, 77, 79, 83, 89, 97, 101, 103, 107, 109, 112, 113, 115, 117, 127, 131, 132, 133, 135, 137, 139, 149, 151, 157, 163, 167, 172, 173, 175, 177, 179

Offset: 1

M. F. Hasler, Nov 02 2009

A number is in this sequence if and only if it is prime or of the form a(k)*10^m+a(n), where a(k), a(n) are in this sequence and 10^m >= a(n) (and from this follows that one among a(k), a(n) can be taken to be prime).
This contains A152242 as a subsequence, but also additional terms like e.g. 202 which can be split into two primes, 2 and 02 (= 2). Such a splitting, where some of the substrings contain leading zeros, is not allowed in A152242.
Terms not in A152242 are listed in A166505.

Henri Picciotto, Selected Integer Sequences

Cf. A152242 (no leading zeros allowed).

Cf. A085823 (super-slimes: all substrings are prime). - Henri Picciotto, Apr 01 2015

is_A166504(n)={ isprime(n) || ((bittest(n,0) || n%10==2) & for(i=1,#Str(n)-1, isprime(n%10^i) & is_A166504(n\10^i) & return(1)))}

is(n)=if(isprime(n),return(1));if(n<202,return(isprime(n%10)&&isprime(n\10)));my(k=n%10,v);if(k==5||k==2,return(if(n<6,1,n\=10;has(n/10^valuation(n,10)))));if(k%2==0,return(0));v=digits(n);for(i=1,#v,if(isprime(n%10^i)&&is(n\10^i),return(1)));0 \\ Charles R Greathouse IV, Apr 30 2013

Edited by Charles R Greathouse IV, Apr 23 2010

Name "Slime numbers", after Henri Picciotto, added by N. J. A. Sloane, Mar 25 2015

A085557 Numbers that have more prime digits than nonprime digits.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232

Offset: 1

Jason Earls, Jul 04 2003

Begins to differ from A046034 at the 21st term (which is the first 3-digit term).

			133 is in the sequence as the prime digits are 3 and 3 (those are two digits; counted with multiplicity) and one nonprime digit 1 and so there are more prime digits than nonprime digits. - _David A. Corneth_, Sep 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A193238, A046034, A046035, A118950, A019546, A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A202267, A202268, A211681.

is(n) = my(d = digits(n), c = 0); for(i = 1, #d, if(isprime(d[i]), c++)); c<<1 > #d \\ David A. Corneth, Sep 06 2020

from itertools import count, islice
def A085557_gen(startvalue=1): # generator of terms
    return filter(lambda n:len(s:=str(n))<(sum(1 for d in s if d in {'2','3','5','7'})<<1),count(max(startvalue,1)))
A085557_list = list(islice(A085557_gen(),20)) # Chai Wah Wu, Feb 08 2023

A202263 Primes in which all substrings and reversal substrings are primes.

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 7 terms.

Subsequence of A085823, A068669, A024770, A012883.

			All substrings and reversal substrings of 373 are primes:3, 7, 37, 73, 373.

Cf. A202264 (noncomposite numbers in which all substrings and reversal substrings are noncomposite), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A213304 Smallest number with n nonprime substrings (Version 3: substrings with leading zeros are counted as nonprime if the corresponding number is not a prime).

Original entry on oeis.org

2, 1, 10, 14, 101, 104, 144, 1001, 1014, 1044, 1444, 10010, 10014, 10144, 10444, 14444, 100120, 100104, 100144, 101444, 104444, 144444, 1000144, 1001040, 1001044, 1001444, 1014444, 1044444, 1444444, 10001044, 10001444, 10010404, 10010444, 10014444, 10144444, 10444444, 14444444, 100010404, 100010444, 100014444, 100104044, 100104444, 100144444, 101444444, 104444444, 144444444

Offset: 0

Hieronymus Fischer, Aug 26 2012

The sequence is well defined since for each n >= 0 there is a number with n nonprime substrings.

Different from A213303, first difference is at a(16).

			a(0)=2, since 2 is the least number with zero nonprime substrings.
a(1)=1, since 1 has 1 nonprime substrings.
a(2)=10, since 10 is the least number with 2 nonprime substrings, these are 1 and 10 ('0' will not be counted).
a(3)=14, since 14 is the least number with 3 nonprime substrings, these are 1 and 4 and 14. 10, 11 and 12 only have 2 such 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(m(m+1)/2) = (13*10^(m-1)-4)/9, m>0.
With b(n):=floor((sqrt(8*n-7)-1)/2):
a(n) > 10^b(n), for n>2, a(n) = 10^b(n) for n=1,2.
a(n) >= 10^b(n)+4*10^(n-1-b(n)(b(n)+1)/2)-1)/9, equality holds if n or n+1 is a triangular number > 0 (cf. A000217).
a(n) >= A213303(n).
a(n) <= A213307(n).

Previous Showing 11-20 of 70 results. Next

cp's OEIS Frontend

A213302 Smallest number with n nonprime substrings (Version 1: substrings with leading zeros are considered to be nonprime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A217109 Minimal number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A202262 Composite numbers in which all substrings are composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A217112 Greatest number (in decimal representation) with n nonprime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A217119 Greatest number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A166504 Slime numbers: numbers which are the concatenation of primes, with "leading zeros" allowed.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Extensions

A085557 Numbers that have more prime digits than nonprime digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A202263 Primes in which all substrings and reversal substrings are primes.