A085823 -id:A085823 - cp's OEIS frontend

A213303 Smallest number with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

2, 1, 10, 14, 101, 104, 144, 1001, 1014, 1044, 1444, 10010, 10014, 10144, 10444, 14444, 100101, 100104, 100144, 101444, 104444, 144444, 1000144, 1001014, 1001044, 1001444, 1014444, 1044444, 1444444, 10001044, 10001444, 10010144, 10010444, 10014444, 10144444, 10444444, 14444444, 100010144

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 A213304, first different term is 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) <= A213304(n).
a(n) <= A213306(n).

A213306 Minimal prime with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

Original entry on oeis.org

2, 13, 11, 103, 101, 149, 1009, 1021, 1049, 1481, 10039, 10069, 10169, 11681, 14669, 100109, 100189, 100169, 101681, 104681, 146669, 1000669, 1001041, 1001081, 1004669, 1014469, 1046849, 1468469, 10001081, 10004669, 10010851

Offset: 0

Hieronymus Fischer, Aug 26 2012

			a(0) = 2, since 2 is the least number with zero nonprime substrings.
a(1) = 13, since 13 has 1 nonprime substring (=’1’).
a(2) = 11, since 11 is the least number with 2 nonprime substrings (= 2 times ‘1’).
a(3) = 103, since 103 is the least number with 3 nonprime substrings, these are ‘1’ and ‘10’ and ‘03’ (‘0’ is not a valid substring in version 2).

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>2.
a(n) >= A213303(n).
a(n) <= A213307(n).

A213307 Minimal prime 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, 13, 11, 127, 101, 149, 1009, 1063, 1049, 1481, 10091, 10069, 10169, 11681, 14669, 100129, 100189, 100169, 101681, 104681, 146669, 1000669, 1001219, 1001081, 1004669, 1014469, 1046849, 1468469, 10001081, 10004669, 10010851

Offset: 0

Hieronymus Fischer, Aug 26 2012

			a(0) = 2, since 2 is the least number with zero nonprime substrings.
a(1) = 13, since 13 there is one nonprime substring (=1).
a(2) = 11, since 11 is the least number with 2 nonprime substrings (2 times ‘1’).
a(3) = 127, since 127 is the least number with 3 nonprime substrings, these are 1 and 12 and 27 (according to version 3).

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>2.
a(n) >= A213304(n).
a(n) >= A213306(n).

A217103 Minimal number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 3, 4, 14, 9, 34, 29, 30, 27, 89, 88, 83, 84, 81, 268, 251, 250, 248, 245, 243, 752, 754, 746, 740, 734, 731, 729, 2237, 2239, 2210, 2203, 2198, 2192, 2189, 2187, 6632, 6611, 6614, 6584, 6577, 6569, 6563, 6564, 6561, 19814, 19754, 19733, 19736, 19706

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} 3^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-3 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-3 representation), p != 1 (mod 3), m>=d, than b := p*3^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_3 is the least number with zero nonprime substrings in base-3 representation.
a(1) = 1, since 1 = 1_3 is the least number with 1 nonprime substring in base-3 representation.
a(2) = 3, since 3 = 10_3 is the least number with 2 nonprime substrings in base-3 representation (0 and 1).
a(3) = 4, since 4 = 11_3 is the least number with 3 nonprime substrings in base-3 representation (1, 1 and 11).
a(4) = 14, since 14 = 112_3 is the least number with 4 nonprime substrings in base-3 representation, these are 1, 1, 11 and 112 (remember, that substrings with leading zeros are considered to be nonprime).
a(7) = 29, since 29 = 1002_3 is the least number with 7 nonprime substrings in base-3 representation, these are 0, 0, 1, 00, 02, 002 and 100 (remember, that substrings with leading zeros are considered to be nonprime, 2_3 = 2, 10_3 = 3 and 1002_3 = 29 are base-3 prime substrings).

Hieronymus Fischer, Table of n, a(n) for n = 0..702

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

a(n) >= 3^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n=1 or n+1 is a triangular number (cf. A000217).
a(n) >= 3^floor((sqrt(8*n+1)-1)/2) for n>3, equality holds if n+1 is a triangular number.
a(A000217(n)-1) = 3^(n-1), n>1.
a(A000217(n)-k) >= 3^(n-1) + k-1, 1<=k<=n, n>1.
a(A000217(n)-k) = 3^(n-1) + p, where p is the minimal number >= 0 such that 3^(n-1) + p, has k prime substrings in base-3 representation, 1<=k<=n, n>1.

A217303 Minimal natural number (in decimal representation) with n prime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 5, 11, 17, 23, 50, 104, 71, 152, 215, 395, 476, 701, 719, 1367, 1934, 1448, 4127, 4121, 4346, 5822, 12302, 12383, 17468, 25505, 32066, 39113, 51749, 91040, 111509, 110798, 117359, 157211, 332396, 334358, 465092, 333791, 819386, 865232, 1001375, 1396673

Offset: 0

Hieronymus Fischer, Nov 22 2012

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=3*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 3^j = 3^n - 1 or m(n)=1, 2, 22, 222, 2222, 22222, …,for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-3 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 3.

			a(1) = 2 = 2_3, since 2 is the least number with 1 prime substring in base-3 representation.
a(2) = 5 = 12_3, since 5 is the least number with 2 prime substrings in base-3 representation (2_3 and 12_3).
a(3) = 11 = 102_3, since 11 is the least number with 3 prime substrings in base-3 representation (2_3, 10_3, and 102_3).
a(5) = 23 = 212_3, since 23 is the least number with 5 prime substrings in base-3 representation (2 times 2_3, 12_3=5, 21_3=19, and 212_3=23).
a(7) = 104 = 10212_3, since 104 is the least number with 7 prime substrings in base-3 representation (2 times 2_3, 10_3=3, 12_3=5, 21_3=19, 102_3=11, and 212_3=23).

Hieronymus Fischer, Table of n, a(n) for n = 0..110

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

a(n) > 3^floor(sqrt(8*n+1)-1)/2), for n>1.
a(n) <= 3^n - 1.
a(n+1) <= 3a(n)+2.

A217308 Minimal natural number (in decimal representation) with n prime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 2, 11, 19, 83, 107, 157, 669, 751, 1259, 4957, 6879, 6011, 14303, 47071, 48093, 65371, 188143, 327515, 440287, 384751, 1029883, 2604783, 2948955, 3602299, 6946651, 20304733, 23846747, 23937003, 23723867, 57278299, 167689071, 175479547, 191496027, 233824091

Offset: 0

Hieronymus Fischer, Nov 22 2012

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=8*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 8^j = 2*(8^n - 1)/7 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-8) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-8 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 8.

			a(1) = 2 = 2_8, since 2 is the least number with 1 prime substring in base-8 representation.
a(2) = 11 = 13_8, since 11 is the least number with 2 prime substrings in base-8 representation (3_8 and 13_8).
a(3) = 19 = 23_8, since 19 is the least number with 3 prime substrings in base-8 representation (2_8, 3_8, and 23_8).
a(4) = 83 = 123_8, since 83 is the least number with 4 prime substrings in base-8 representation (2_8, 3_8, 23_8=19, and 123_8=83).
a(8) = 751 = 1357_8, since 751 is the least number with 8 prime substrings in base-8 representation (3_8, 5_8, 7_8, 13_8=11, 35_8=29, 57_8=47, 357_8=239, and 1357_8=751).

Hieronymus Fischer, Table of n, a(n) for n = 0..45

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

a(n) > 8^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= 2*(8^n - 1)/7, n>0.
a(n+1) <= 8*a(n)+2.

A085822 Smallest prime with n prime substrings (excluding prime itself but allowing leading zeros).

Original entry on oeis.org

2, 13, 23, 113, 137, 373, 1973, 1733, 1373, 10337, 10313, 31379, 37337, 113173, 211373, 313739, 337397, 1113173, 1003733, 2313797, 2337397, 10003733, 11031373, 11373379, 33133733, 20037337, 100037339, 103137337

Offset: 0

Zak Seidov, Jul 04 2003

			a(5)=373 because it is prime and has 5 prime substrings: 3,7,3,37,73.

Cf. A085823.

V:= Array(0..27): count:= 0:
p:= 1:
while count < 28 do
  p:= nextprime(p);
  v:= nps(p);
  if v <= 100 and V[v] = 0 then V[v]:= p; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Mar 03 2023

More terms from Robert Israel, Mar 03 2023

A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 113, 131, 311, 313, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 17 terms.
Supersequence of A202263, A085823.
Subsequence of A068669, A012883, A024770, A012883.

			All substrings and reversal substrings of 311 are noncomposites: 1, 3, 11, 13, 31, 113, 311.

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A213299 Partial sums of A211681.

Original entry on oeis.org

2, 5, 10, 17, 40, 77, 130, 203, 440, 813, 1350, 2087, 4460, 8197, 13570, 20943, 44680, 82053, 135790, 209527, 446900, 820637, 1358010, 2095383, 4469120, 8206493, 13580230, 20953967, 44691340

Offset: 1

Hieronymus Fischer, Jun 08 2012

The terms are primes for n = 1, 2, 4, 12, 22, 32 and possibly further n’s (Question).

Hieronymus Fischer, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,9,-18,9,0,10,-20,10).

Cf. A019546, A035232, A039996, A046034, A085823, A211681, A211682.

a(n) = ((3982 + 2709*k + 567*k^2 + 54*k^3)*10^m - 1980*m - 2200 - 495*k + 162*((n+1) mod 2) * (-1)^m * (-1)^floor(n/2))/891, where m=floor((n-1)/4), k=(n-1) mod 4.
G.f.: (2*x*(1+x^10) + 3*x^2*(1 + x^3 + x^5 + x^6) + 5*x^3*(1+x^6) + 7*x^4*(1+x^2))/((1-x)*(1-10*x^4)*(1-x^8)).
From Chai Wah Wu, Feb 08 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 9*a(n-4) - 18*a(n-5) + 9*a(n-6) + 10*a(n-8) - 20*a(n-9) + 10*a(n-10) for n > 10.
G.f.: x*(-2*x^7 + 2*x^6 - 5*x^5 + 2*x^4 - 2*x^3 - 2*x^2 - x - 2)/((x - 1)^2*(x^4 + 1)*(10*x^4 - 1)). (End)

Typo in g.f. corrected by Hieronymus Fischer, Sep 03 2012

A213301 Largest prime with n nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

373, 3797, 37337, 37397, 373379, 831373, 973373, 3733739, 8313733, 9973331, 9721373, 52313797, 73313797, 97337333, 99793373, 373373977, 831373379, 799733317, 974313797, 991733137, 7331337337, 3797193373, 9719337973, 9917331373, 9793733797, 9974331373

Offset: 0

Hieronymus Fischer, Aug 26 2012

			a(0)=373, since 373 is the greatest prime such that all substrings are prime, hence it is the maximal prime with 0 nonprime substrings.
a(3)= 37397, since the nonprime substrings of 37337 are 9, 39 and 7397, and all greater primes have > 3 nonprime substrings.

Hieronymus Fischer, Table of n, a(n) for n = 0..25

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

Cf. A035244, A079307, A213300 - A213321.

cp's OEIS Frontend

A213303 Smallest number with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

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A213306 Minimal prime with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

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A213307 Minimal prime with n nonprime substrings (Version 3: substrings with leading zeros are counted as nonprime if the corresponding number is not a prime).

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A217103 Minimal number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

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A217303 Minimal natural number (in decimal representation) with n prime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

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A217308 Minimal natural number (in decimal representation) with n prime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

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A085822 Smallest prime with n prime substrings (excluding prime itself but allowing leading zeros).

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A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

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A213299 Partial sums of A211681.

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A213301 Largest prime with n nonprime substrings (substrings with leading zeros are considered to be nonprime).

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