A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

-1, -1, 0, 0, 1, 0, -1, 0, -1, -1, 1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 1, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 7, 0, 7, -1, -1, 4, 15, 0, -1, 12, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1, 0

Offset: 0

Author

Chai Wah Wu, Feb 23 2021

Keywords

• sign
• base

Comments

A variation of A341716. a(n) = n-1 for n = 82. Are there other n such that a(n) = n-1?

Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

Examples

			a(10) = 1 since 109 is prime. a(22) = 1 since 2221 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A052088, A052089, A054211, A341701, A341715, A341716, A341717.

Programs

Maple

tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
f:= proc(n) local x,k;
  x:= n;
  for k from 0 to n-1 do
    if isprime(x) then return k fi;
    x:= tcat(x,n-k-1)
  od;
  -1
end proc:
map(f, [$0..100]); # Robert Israel, Mar 02 2022

PARI

a(n) = my(k=0, s=Str(n)); while (!isprime(eval(s)), k++; n--; if (k>=n, return(-1)); s = concat(s, Str(n-k))); return(k); \\ Michel Marcus, Mar 02 2022

Python

from sympy import isprime
def A341702(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return n-m-1 if isprime(k) else -1

Formula

a(n) = n-A341701(n).

a(p) = 0 if and only if p is prime.