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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.

%I A341702 #20 Mar 02 2022 12:08:42
%S A341702 -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,
%T A341702 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,
%U A341702 -1,12,9,-1,1,0,13,0,-1,-1,-1,-1,-1,0,57,-1,1,0,-1,0
%N 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.
%C A341702 A variation of A341716. a(n) = n-1 for n = 82. Are there other n such that a(n) = n-1?
%C A341702 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).
%H A341702 Robert Israel, <a href="/A341702/b341702.txt">Table of n, a(n) for n = 0..2000</a>
%F A341702 a(n) = n-A341701(n).
%F A341702 a(p) = 0 if and only if p is prime.
%e A341702 a(10) = 1 since 109 is prime. a(22) = 1 since 2221 is prime.
%p A341702 tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
%p A341702 f:= proc(n) local x,k;
%p A341702   x:= n;
%p A341702   for k from 0 to n-1 do
%p A341702     if isprime(x) then return k fi;
%p A341702     x:= tcat(x,n-k-1)
%p A341702   od;
%p A341702   -1
%p A341702 end proc:
%p A341702 map(f, [$0..100]); # _Robert Israel_, Mar 02 2022
%o A341702 (Python)
%o A341702 from sympy import isprime
%o A341702 def A341702(n):
%o A341702     k, m = n, n-1
%o A341702     while not isprime(k) and m > 0:
%o A341702         k = int(str(k)+str(m))
%o A341702         m -= 1
%o A341702     return n-m-1 if isprime(k) else -1
%o A341702 (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
%Y A341702 Cf. A052088, A052089, A054211, A341701, A341715, A341716, A341717.
%K A341702 sign,base
%O A341702 0,26
%A A341702 _Chai Wah Wu_, Feb 23 2021