A030670 Smallest prime formed by appending a number to the n-th prime.

23, 31, 53, 71, 113, 131, 173, 191, 233, 293, 311, 373, 419, 431, 479, 5323, 593, 613, 673, 719, 733, 797, 839, 8923, 971, 1013, 1031, 10711, 1091, 11311, 1277, 1319, 1373, 1399, 1493, 1511, 1571, 1637, 16729, 1733, 17911, 1811, 1913, 1931

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base

Comments

Previous name: Smallest prime whose decimal expansion begins (nontrivially) with the n-th prime.

Add digits to p (starting with a nonzero digit) until another prime is reached.

This differs from A064792 in that there the appended digits may start with a 0. The first difference occurs at a(16) = 5323, while A064792(16) = 5303. - M. F. Hasler, Jan 15 2025

Examples

			a(16) = 5323 because 53 is the 16th prime, and 23 is the smallest number that can be appended to 53 to give another prime. 5303 is not allowed because 03 starts with zero. - _David Radcliffe_, Jan 08 2025
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

See A064792 for another version. Note that A064792 <= a(n). Cf. A065112, A178220.

Programs

Maple

f:= proc(p) local d,x;
  for d from 1 do
    x:= nextprime(10^d*p+10^(d-1)-1);
    if x < 10^d*(p+1) then return x fi
  od
end proc:
map(f @ ithprime, [$1..100]); # Robert Israel, Aug 12 2018

Mathematica

f[n_] := Block[{k = 1, p = Prime@ n}, While[a = 10^Floor[1 + Log10@ k] p + k; !PrimeQ@ a, k += 2]; a]; Array[f, 44]

PARI

apply( {A030670(n)=n=prime(n);for(L=1,oo, n*=10; forstep(s=bitor(10^(L-1),1),10^L-1,2, isprime(n+s)&& return(n+s)))}, [1..44]) \\ M. F. Hasler, Jan 15 2025

Python

from sympy import prime, isprime
from itertools import count
def a030670(n):
  p = str(prime(n))
  return next(x for k in count(1) if isprime(x:=int(p+str(k)))) # David Radcliffe, Jan 08 2025

Extensions

Title changed by David Radcliffe, Jan 08 2025