A046034 -id:A046034 - 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.

A276729 Number of nonprime digits in the decimal expansion of n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2

Offset: 0

M. F. Hasler, Sep 16 2016

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A055642, A193238, A085557, A046034 (indices of 0's), A084984, A085558, A202267, A202268.

Cf. A011557 (integers that give records).

f:= proc(n) local t; option remember;
  t:= n mod 10;
procname((n-t)/10) + `if`(member(t,[2,3,5,7]),0,1)
end proc:
f(0):= 0:
1,seq(f(i),i=1..100); # Robert Israel, Feb 27 2019

PARI
```
a(n)=#select(t->!isprime(t),digits(n))
```

a(n) = A055642(n) - A193238(n).

a(A046034(n)) = 0. - Gordon Atkinson, Sep 06 2019

A374665 a(n) is the first number that is the concatenation of n primes and also the product of n primes (counted with multiplicity).

Original entry on oeis.org

2, 22, 222, 2223, 22232, 222222, 2222325, 22222272, 222225552, 2222223255, 22222335232, 222222327525, 2222222372352, 22222222575552, 222222223327232, 2222222225252352, 22222222223327232, 222222222272535552, 2222222222225252352, 22222222222327775232, 222222222222737375232

Offset: 1

Robert Israel, Jul 15 2024

Is this a subsequence of A046034?

			a(5) = 22232 because 22232 is the concatenation of the 5 primes 2, 2, 2, 3, 2 and 22232 = 2^3 * 7 * 397 is the product of 5 primes (counted with multiplicity).

Cf. A001222, A046034. Main diagonal of A374376.

f:= proc(n) local k, x, W, L, i;
  W:= [2,3,5,7];
  for k from 0 to 4^n-1 do
    L:= convert(4^n+k,base,4);
    x:= add(W[L[i]+1]*10^(i-1),i=1..n);
    if numtheory:-bigomega(x) = n then return x fi;
  od;
end proc:
map(f, [$1..20]);

A001222(a(n)) = n.

A087368 Prime-indexed primes (PIPs) whose digits are all primes.

Original entry on oeis.org

3, 5, 277, 353, 773, 3733, 5557, 7523, 7753, 25357, 25733, 27733, 32233, 32323, 32533, 37273, 53233, 53353, 53377, 53777, 55733, 72337, 72727, 73757, 77377, 77557, 232523, 272333, 275773, 322727, 327553, 327757, 333233, 352357, 353527

Offset: 1

Cino Hilliard, Oct 21 2003

Chances are these numbers are infinite since PIPs are infinite.

			59 is prime, the 59th prime is 277, and 2 and 7 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..14683 (terms below 10^11)
Albert Frank and Philippe Jacqueroux, International Contest, 2001. Item 12.

Intersection of A006450 and A046034.

Select[Flatten[Table[FromDigits /@ Tuples[Prime[Range[4]], k], {k, 1, 6}]], PrimeQ[#] && PrimeQ[PrimePi[#]] &] (* Amiram Eldar, Jul 08 2024 *)

pip(n) = { for(x=1,n, flag=1; y=prime(prime(x)); y2=y; for(j=1,length(Str(y)), r = y%10; if(!isprime(r),flag=0); y = floor(y/10); ); if(flag,print1(y2",")); ) }

Offset corrected by Amiram Eldar, Jul 08 2024

A153025 Numbers n with property that n and n^2 use only prime digits.

Original entry on oeis.org

5, 235, 72335

Offset: 1

Zak Seidov, Dec 17 2008

Probably there are no other terms. No other terms up to 10^40.

Intersection of A046034 and A275971. - M. F. Hasler, Sep 16 2016

			The squares of 5, 235, 72335 are 25, 55225, 5232352225.

Cf. A046034, A275971.

Flatten[Table[Select[FromDigits/@Tuples[{2,3,5,7},n],AllTrue[ IntegerDigits[ #^2], PrimeQ]&],{n,5}]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 16 2014 *)

Edited by David Wilson and N. J. A. Sloane, Jan 25 2009

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A276729 Number of nonprime digits in the decimal expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A374665 a(n) is the first number that is the concatenation of n primes and also the product of n primes (counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A087368 Prime-indexed primes (PIPs) whose digits are all primes.

Views

Author

Keywords

Comments

Examples

Links