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

1,1

Comments

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

Links

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]

Example

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

Maple

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

Mathematica

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}]

Prog

(PARI) {s=Vec(Str(c=1)); for(d=1, 30, for(j=1, 9e9,
#s<d+j && s=concat( s, Vec( Str( c++ ))); s[j]=="0" && next;
isprime( p=eval( concat( vecextract( s, Str(j, "..", j+d-1) )))) || next;
print(d, " ", p); next(2)))} /* replace "isprime" by 2==bigomega to get the semiprime analog */ \\ M. F. Hasler, Aug 23 2008

Crossrefs

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

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

Sequence in context: A131176 A141405 A068876 * A352164 A141888 A285811

Adjacent sequences: A073172 A073173 A073174 * A073176 A073177 A073178

Keyword

base,nonn

Author

Zak Seidov, Aug 22 2002

Extensions

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

Status

approved