A003617 -id:A003617 - cp's OEIS frontend

A073175 First occurrence of an n-digit prime as a substring in the concatenation of the natural numbers 12345678910111213141516171819202122232425262728293031....

2, 23, 101, 4567, 67891, 789101, 4567891, 23456789, 728293031, 1234567891, 45678910111, 678910111213, 1222324252627, 12345678910111, 415161718192021, 3637383940414243, 12223242526272829, 910111213141516171

Offset: 1

Zak Seidov, Aug 22 2002

This is to Champernowne's constant 0.12345678910111213... (Sloane's A033307) as A073062 is to A033308 Decimal expansion of Copeland-Erdos constant: concatenate primes. - Jonathan Vos Post, Aug 25 2008

			Take 1234567891011121314151617....; a(4)=4567 because the first 4-digit prime in the sequence is 4567.
1213 is < 4567 but occurs later in the string.
a(5) = 67891 is the first occurrence of a five-digit substring that is a prime, 12345(67891)011121314...
a(1) = 2 = prime(1). a(2) = 23 = prime(9). a(3) = 571 = prime(105). a(4) = 2357 = prime(350). a(5) = 11131 = prime(1349). - _Jonathan Vos Post_, Aug 25 2008

Robert Israel, Table of n, a(n) for n = 1..999
Eric W. Weisstein, Champernowne Constant. [From _Jonathan Vos Post_, Aug 25 2008]
Eric W. Weisstein, Copeland-Erdos Constant. [From _Jonathan Vos Post_, Aug 25 2008]

Cf. A003617. - M. F. Hasler, Aug 23 2008

Cf. A000040, A033307, A033308, A073062. - Jonathan Vos Post, Aug 25 2008

N:= 1000: # to use the concatenation of 1 to N
L:= NULL:
for n from 1 to N do
  L:= L, op(ListTools:-Reverse(convert(n,base,10)))
od:
L:= [L]:
nL:= nops(L);
f:= proc(n) local k,B,x;
  for k from 1 to nL-n+1 do
    B:= L[k..k+n-1];
    x:= add(B[i]*10^(n-i),i=1..n);
    if isprime(x) then return x fi
  od;
false;
end proc:
seq(f(n),n=1..100); # Robert Israel, Aug 16 2018

p200=Flatten[IntegerDigits[Range[200]]]; Do[pn=Partition[p200, n, 1]; ln=Length[pn]; tab=Table[Sum[10^(n-k)*pn[[i, k]], {k, n}], {i, ln}]; Print[{n, Select[tab, PrimeQ][[1]]}], {n, 20}]

{s=Vec(Str(c=1)); for(d=1,30, for(j=1,9e9,
#sM. F. Hasler, Aug 23 2008

Edited by N. J. A. Sloane, Aug 19 2008 at the suggestion of R. J. Mathar

A131581 The next prime greater than the square root of 10^n.

Original entry on oeis.org

2, 5, 11, 37, 101, 317, 1009, 3163, 10007, 31627, 100003, 316241, 1000003, 3162283, 10000019, 31622777, 100000007, 316227767, 1000000007, 3162277669, 10000000019, 31622776621, 100000000003, 316227766069, 1000000000039

Offset: 0

Robert G. Wilson v, Jan 12 2007

The difference between a(n) and floor(sqrt(10^n)): 1, 2, 1, 6, 1, 1, 9, 1, 7, 5, 3, 14, 3, 6, 19, 1, 7, 1, 7, 9, ....

Values for which the difference between a(n) and floor(sqrt(10^n)) equals one: 0, 2, 4, 5, 7, 15, 17, 25, 145, 1079, ..., (1350). Only even terms are 0, 2 & 4; all the rest must be odd.

Cf. A003617, A132153, A136582, A175733, A175734.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[ NextPrim@ Floor@ Sqrt[10^n], {n, 0, 25}]
Table[NextPrime[Sqrt[10^n]],{n,0,30}] (* Harvey P. Dale, Aug 15 2017 *)

a(n) = sqrt(A175733(n+1)). [From Jaroslav Krizek, Aug 24 2010]

A098449 Smallest n-digit semiprime.

Original entry on oeis.org

4, 10, 106, 1003, 10001, 100001, 1000001, 10000001, 100000001, 1000000006, 10000000003, 100000000007, 1000000000007, 10000000000015, 100000000000013, 1000000000000003, 10000000000000003, 100000000000000015, 1000000000000000007, 10000000000000000001

Offset: 1

Rick L. Shepherd, Sep 07 2004

Derek Orr, Table of n, a(n) for n = 1..60

Cf. A098450 (largest n-digit semiprime), A003617 (smallest n-digit prime), A001358 (semiprimes).

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := NextSemiPrime[10^n - 1]; Array[f, 19, 0] (* Robert G. Wilson v, Dec 18 2012 *)
Table[Module[{k=0},While[PrimeOmega[10^n+k]!=2,k++];10^n+k],{n,0,20}] (* Harvey P. Dale, Jun 15 2025 *)

a(n)=for(k=10^(n-1),10^n-1,if(bigomega(k)==2,return(k)))
vector(50, n, a(n)) \\ Derek Orr, Aug 15 2014

from sympy import factorint
def semiprime(n): f = factorint(n); return sum(f[p] for p in f) == 2
def a(n):
  an = max(1, 10**(n-1))
  while not semiprime(an): an += 1
  return an
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 10 2021

A064490 Smallest prime with prime(n) decimal digits.

Original entry on oeis.org

11, 101, 10007, 1000003, 10000000019, 1000000000039, 10000000000000061, 1000000000000000003, 10000000000000000000009, 10000000000000000000000000331, 1000000000000000000000000000057, 1000000000000000000000000000000000067, 10000000000000000000000000000000000000121

Offset: 1

Jason Earls, Oct 04 2001

			11 is the first prime with 2 decimal digits.
101 is the first prime with 3 decimal digits.

Alois P. Heinz, Table of n, a(n) for n = 1..168 (first 75 terms from Harry J. Smith)

Cf. A000040, A003617, A064489.

for n from 1 to 20 do p := ithprime(n): for i from 10^(p-1) to 10^p do if isprime(i) then printf(`%d,`,i); break; fi: od: od:
# second Maple program:
a:= n-> nextprime(10^(ithprime(n)-1)):
seq(a(n), n=1..15);  # Alois P. Heinz, Jun 24 2018

Table[NextPrime[10^(n-1)],{n,Prime[Range[15]]}] (* Harvey P. Dale, Feb 06 2020 *)

l(n)=ln=0; while(n,n=floor(n/10); ln++); return(ln);
a=0; for(n=1,10^6,x=l(prime(n)); if(isprime(x),b=x; if(b>a,a=b; print(prime(n)))))

{ for (n=1, 75, p=prime(n); a=nextprime(10^(p - 1)); write("b064490.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 16 2009

from sympy import prime, nextprime, primepi
def a(n): return nextprime(10**(prime(n)-1))
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, May 26 2021

a(n) = A003617(A000040(n)).

a(n) = A000040(A064489(n)).

More terms from James Sellers, Oct 08 2001

Offset changed from 0 to 1 by Harry J. Smith, Sep 16 2009

A069100 n-th n-digit prime number.

Original entry on oeis.org

2, 13, 107, 1021, 10061, 100069, 1000117, 10000189, 100000193, 1000000181, 10000000259, 100000000193, 1000000000303, 10000000000411, 100000000000487, 1000000000000513, 10000000000000931, 100000000000000591, 1000000000000000861, 10000000000000000667

Offset: 1

Darrell Minor, Apr 05 2002

Eduard Roure Perdices, Table of n, a(n) for n = 1..1000

Cf. A003617, A107108, A107109.

a[n_] := NextPrime[10^(n - 1), n]; (* Eduard Roure Perdices, May 17 2021 *)

a(n)=prime(primepi(10^(n-1))+n) \\ Franklin T. Adams-Watters, Mar 07 2014

from sympy import nextprime
def a(n):  return nextprime(10**(n-1), ith=n)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 16 2021

More terms from Vladeta Jovovic, Apr 07 2002

Extended by Eduard Roure Perdices, May 17 2021

A090226 Least k such that prime(k) has n digits. Index of least n-digit prime.

Original entry on oeis.org

1, 5, 26, 169, 1230, 9593, 78499, 664580, 5761456, 50847535, 455052512, 4118054814, 37607912019, 346065536840, 3204941750803, 29844570422670, 279238341033926, 2623557157654234, 24739954287740861, 234057667276344608, 2220819602560918841, 21127269486018731929

Offset: 1

Amarnath Murthy, Nov 25 2003

Eduard Roure Perdices, Table of n, a(n) for n = 1..29
C. Caldwell's Prime Pages, A Table of values of pi(x).
X. Gourdon, A Table of values of pi(x).

Cf. A003617 (values of prime(k)), A006880.

Table[PrimePi[NextPrime[10^n]],{n,0,14}] (* This generates the first 15 terms of the sequence, but Mathematica cannot generate the 16th term using this program. *) (* Harvey P. Dale, May 12 2019 *)

a(1)=1; a(n) = pi(10^(n-1)) + 1 for n > 1 where pi(x) is the number of primes less than x. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 26 2004

a(n) = A006880(n-1) + 1, for n >= 1. - Eduard Roure Perdices, Apr 18 2021

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 26 2004

Extended by Eduard Roure Perdices, Apr 18 2021

A139053 Array read by rows: row n lists the first 3 primes with n digits.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 101, 103, 107, 1009, 1013, 1019, 10007, 10009, 10037, 100003, 100019, 100043, 1000003, 1000033, 1000037, 10000019, 10000079, 10000103, 100000007, 100000037, 100000039, 1000000007, 1000000009, 1000000021, 10000000019, 10000000033, 10000000061

Offset: 1

Omar E. Pol, Apr 08 2008

Michael S. Branicky, Table of n, a(n) for n = 1..3000

Cf. A000040, A003617, A073914, A139052, A139054.

np[n_]:=Module[{a=NextPrime[10^n]},{a,NextPrime[a], NextPrime[NextPrime[ a]]}]; Flatten[Array[np,12,0]] (* Harvey P. Dale, Dec 14 2011 *)
Flatten@Array[NextPrime[10^#,{1,2,3}]&,12,0] (* Giorgos Kalogeropoulos, May 06 2019 *)

from sympy import nextprime
def auptodigs(maxdigits):
  alst = []
  for n in range(1, maxdigits+1):
    p1 = nextprime(10**(n-1))
    p2 = nextprime(p1)
    p3 = nextprime(p2)
    alst.extend([p1, p2, p3])
  return alst
print(auptodigs(11)) # Michael S. Branicky, May 07 2021

More terms from Max Alekseyev, Dec 12 2011

a(32) and beyond from Michael S. Branicky, May 07 2021

A139054 Array read by rows: row n lists the first 4 primes with n digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 101, 103, 107, 109, 1009, 1013, 1019, 1021, 10007, 10009, 10037, 10039, 100003, 100019, 100043, 100049, 1000003, 1000033, 1000037, 1000039, 10000019, 10000079, 10000103, 10000121, 100000007, 100000037, 100000039, 100000049

Offset: 1

Omar E. Pol, Apr 08 2008

First differs from A000040 at a(9). - Omar E. Pol, Feb 13 2014

			Array begins:
    2,   3,   5,   7;
   11,  13,  17,  19;
  101, 103, 107, 109;
  ...

Cf. A000040, A003617, A073914, A139052, A139053.

Flatten @ Table[ NextPrime[10^i, Range @ 4], {i, 0, 9}] (* Mikk Heidemaa, Jan 08 2023 *)

a(4n-3) = A003617(n). - Omar E. Pol, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A180922 Smallest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

Original entry on oeis.org

8, 12, 102, 1001, 10002, 100006, 1000002, 10000005, 100000006, 1000000003, 10000000001, 100000000006, 1000000000001, 10000000000001, 100000000000018, 1000000000000002, 10000000000000006, 100000000000000007, 1000000000000000001, 10000000000000000007

Offset: 1

Jonathan Vos Post, Jan 23 2011

This is to 3 as smallest n-digit semiprime A098449 is to 2, and as smallest n-digit prime A003617 is to 1. Smallest n-digit triprime. Smallest n-digit 3-almost prime.

			a(1) = 8 because 8=2^3 is the smallest (only) 1-digit number divisible by exactly 3 primes (counted with multiplicity).
a(2) = 12 because 12 = 2^2 * 3 is the smallest of the (21) 2-digit numbers divisible by exactly 3 primes (counted with multiplicity).
a(3) = 102 because 102 = 2 * 3 * 17 is the smallest 3-digit numbers divisible by exactly 3 primes (counted with multiplicity).

Cf. A003617, A014612, A098449.

A180922(n)=for(n=10^(n-1),10^n-1,bigomega(n)==3&return(n)) \\ M. F. Hasler, Jan 23 2011

from sympy import factorint
def triprimes(n): f = factorint(n); return sum(f[p] for p in f) == 3
def a(n):
  an = max(1, 10**(n-1))
  while not triprimes(an): an += 1
  return an
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 10 2021

A249447 Least n-digit prime whose digit sum is also prime.

Original entry on oeis.org

2, 11, 101, 1013, 10037, 100019, 1000033, 10000019, 100000037, 1000000033, 10000000019, 100000000019, 1000000000039, 10000000000037, 100000000000031, 1000000000000037, 10000000000000079, 100000000000000013, 1000000000000000031, 10000000000000000051, 100000000000000000039

Offset: 1

Paolo P. Lava, Oct 29 2014

Subsequence of A046704 (primes with digits sum being prime).

Some terms of this sequence are also in A003617, the least n-digit primes. - Michel Marcus, Oct 30 2014

			a(1) = 2 because it is the least prime with just one digit.
a(2) = 11 because it is the least prime with 2 digits whose sum, 1 + 1 = 2, is a prime.
Again, a(7) = 1000033 because it is the least prime with 7 digits whose sum is a prime: 1 + 0 + 0 + 0 + 0 + 3 + 3 = 7.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A003617, A046704, A068166, A249448.

P:=proc(q) local a,b,k,n; for k from 0 to q do
for n from 10^k to 10^(k+1)-1 do if isprime(n) then a:=n; b:=0;
while a>0 do b:=b+(a mod 10); a:=trunc(a/10); od;
if isprime(b) then print(n); break; fi; fi;
od; od; end: P(10^3);

a(n) = {p = nextprime(10^(n-1)); while (!isprime(sumdigits(p)), p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 29 2014

cp's OEIS Frontend

A073175 First occurrence of an n-digit prime as a substring in the concatenation of the natural numbers 12345678910111213141516171819202122232425262728293031....

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A131581 The next prime greater than the square root of 10^n.

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A098449 Smallest n-digit semiprime.

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A064490 Smallest prime with prime(n) decimal digits.

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A069100 n-th n-digit prime number.

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A090226 Least k such that prime(k) has n digits. Index of least n-digit prime.

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Mathematica

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A139053 Array read by rows: row n lists the first 3 primes with n digits.

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A139054 Array read by rows: row n lists the first 4 primes with n digits.