A090465 Smallest number m such that (n+1)*10^m-1 (i.e., n with m nines appended) yields a prime, or -1 if this will always yield a composite number.

1, 0, 0, 2, 0, -1, 0, 1, -1, 1, 0, -1, 0, 1, -1, 2, 0, -1, 0, 2, -1, 1, 0, -1, 3, 1, -1, 4, 0, -1, 0, 2, -1, 1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, -1, 16, 0, -1, 1, 1, -1, 3, 0, -1, 5, 1, -1, 15, 0, -1, 0, 2, -1, 12, 1, -1, 0, 2, -1, 1, 0, -1, 0, 2, -1, 1, 3, -1, 0, 1, -1, 1, 0, -1, 1, 2, -1, 33, 0, -1, 1, 1, -1, 3, 10, -1, 0, 3, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1

Offset: 1

Author

Chuck Seggelin, Dec 02 2003

Keywords

• base
• sign

Comments

The first 9 record holders in this sequence are 1, 4, 25, 28, 46, 88, 374, 416, 466 with the values 1, 2, 3, 4, 16, 33, 57, 70, 203 respectively.

The next 3 record holders are 1342, 1802, 1934 with the values 29711, 45882, 51836 respectively. 4420 may be the next record holder as no solution has been found for it yet. 4420 was tested out to 300000 nines with no prime formed. - Toshitaka Suzuki, May 27 2024

Examples

			a(25) = 3 because three 9's must be appended to 25 before a prime is formed (25999).
a(6) = -1 because no matter how many 9's are appended to 6, the resulting number is always divisible by 3 and can therefore not be prime.
		

Links

• Toshitaka Suzuki, Table of n, a(n) for n = 1..4419

Crossrefs

Cf. A083747 (The Wilde Primes, i.e. same operation using ones), A090584 (using threes), A090464 (using sevens).

Programs

Maple

f:= proc(n) local x,m;
  if n mod 3 = 0 and n <> 3 then return -1 fi;
  x:= n;
  for m from 0 to 10^4 do
   if isprime(x) then return m fi;
    x:= 10*x+9
  od;
fail
end proc:
map(f, [$1..200]); # Robert Israel, Jun 05 2024

PARI

apply( {A090465(n, LIM=500)=n%3 && for(m=0, LIM, ispseudoprime(n) && return(m); n=n*10+9); -(n>3)}, [1..55]) \\ Retun value -1 means that a(n) = -1 or, if n%3 > 0, then possibly a(n) > LIM, the search limit given as second (optional) parameter. - M. F. Hasler, Jun 05 2024

Formula

a(p) = 0 for p prime.

a(n) = -1 if n is a proper multiple of 3.

Extensions

Definition edited by M. F. Hasler, Jun 05 2024