A030665 Smallest nontrivial extension of n which is prime.
11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 2003, 211, 223, 233, 241, 251, 263, 271, 281, 293, 307, 311, 3203, 331, 347, 353, 367, 373, 383, 397, 401, 419, 421, 431, 443, 457, 461, 479, 487, 491, 503, 5101
Offset: 1
Examples
For n = 1, we could append 1, 3, 7, 9, 01, etc., to make a prime, but 1 gives the smallest of these, 11, so a(1) = 11. For n = 2, although 2 is already prime, the definition requires an appending at least one digit. 1 doesn't work because 21 = 3 * 7, but 3 does because 23 is prime. Hence a(2) = 23.
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..10000 [T. D. Noe computed the first 1000 terms]
- Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.
- Index entries for primes involving decimal expansion of n
Programs
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Maple
f:= proc(n) local x0, d, r, y; for d from 1 do x0:= n*10^d; for r from 1 to 10^d-1 by 2 do if isprime(x0+r) then return(x0+r) fi od od end proc: seq(f(n), n=1..100); # Robert Israel, Dec 23 2014 -
Mathematica
A030665[n_] := Module[{d = 10, nd = 10 * n}, While[True, x = NextPrime[nd]; If[x < nd + d, Return[x]]; d *= 10; nd *= 10]]; Array[A030665, 100] (* Jean-François Alcover, Oct 19 2016, translated from Chai Wah Wu's Python code *)
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PARI
apply( {A030665(n)=my(p,L); until(n+10^L++>p=nextprime(n), n*=10);p}, [1..55]) \\ M. F. Hasler, Jan 27 2025 -
Python
from sympy import nextprime def A030665(n): d, nd = 10, 10*n while True: x = nextprime(nd) if x < nd+d: return int(x) d *= 10 nd *= 10 # Chai Wah Wu, May 24 2016
Extensions
Corrected by Ray Chandler, Aug 11 2003
Comments