A186070 a(n) is the smallest prefix such that the numbers with k digits "9" appended are primes for k = 1, 2, ..., n.

1, 1, 1, 13, 13, 608, 4094, 1875397, 143639306, 5613099946, 20207317759, 1474035260669

Offset

1,4

Comments

See A186069 for the digit "3" case. The corresponding sequences with the digits "1" or "7" are not possible because if nX and nXX are prime, then nXXX will be a multiple of 3 when X is 1 or 7.

323399992 is prime if you add up to eight "9"s to it. This one is noteworthy since it contains a string of four "9"s to begin with. It also only contains three unique digits. - Jonathan Pappas, Oct 13 2021

Any term after a(7) is congruent to 6 (mod 7). - Jonathan Pappas, Oct 19 2021

When a'(n) is the smallest prefix as in the Name but not for k = n+1, then the data becomes: 2, 5, 1, 104, 13, 608, 4094, ... In this case, a'(2) = 5 because 59 and 599 are primes while 5999 = 7*857. - Bernard Schott, Nov 19 2021

Links

Table of n, a(n) for n=1..12.

Example

a(6) = 608 because 6089, 60899, 608999, 6089999, 60899999 and 608999999 are primes.

Maple

with(numtheory): for n from 1 to 10 do: idd:=0:for k from 1 to 1000000 while(idd=0)
  do:a0:=k:id:=0:ite:=0:for u from 1 to n do:a1:=a0*10+9:if type(a1, prime)=true
  then ite:=ite+1:a0:=a1:else fi:od:if ite =n then idd:=1:print(k):else fi:od:od:

Mathematica

m=1; Table[While[d=IntegerDigits[m]; k=0; While[k++; AppendTo[d, 9]; k <= n
  && PrimeQ[FromDigits[d]]]; k <= n, m++]; m, {n, 8}]

Prog

(PARI) isok(k, n) = my(sj=Str(k)); for(j=1, n, if (!isprime(eval(sj=concat(sj, Str(9)))), return(0))); return(1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 18 2021
(Python)
from sympy import isprime
def a(n):
    prefix = 1
    while not all(isprime(int(str(prefix) + "9"*k)) for k in range(1, n+1)):
        prefix += 1
    return prefix
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Nov 19 2021

Crossrefs

Cf. A185682, A185684, A185685, A185687, A186069.

Sequence in context: A165834 A020553 A094461 * A254069 A309990 A196461

Adjacent sequences: A186067 A186068 A186069 * A186071 A186072 A186073

Keyword

nonn,base,hard,more

Author

Michel Lagneau, Feb 11 2011

Extensions

a(9) from Jonathan Pappas, Oct 13 2021

a(10)-a(11) from Jonathan Pappas, Oct 19 2021

a(12) from Jonathan Pappas, July 13 2023

Status

approved