A186069 a(n) is the smallest prefix such that the numbers with k digits "3" appended are primes for k = 1..n.

1, 2, 2, 2, 2177, 16109, 1100318, 1315351, 74810500, 1130720467, 103273582897, 1587865798465

Offset

1,2

Comments

See A186070 for the digit "9" 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.

Any term after a(7) is congruent to 2 (mod 7). - Jonathan Pappas, Oct 17 2021

a(13) is greater than 3*10^12. - 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: 1, 26, 17, 2, 2177, 16109, ... In this case, a'(2) = 26 because 263 and 2633 are primes, while 26333 is divisible by 17. - Bernard Schott, Nov 18 2021

Links

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

Example

a(4) = 2 because 23, 233, 2333, 23333 are primes and 133 is not a prime number.

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+3: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, 3]; k <= n && PrimeQ[FromDigits[d]]]; k <= n, m++]; m, {n, 6}]

Prog

(PARI) isok(k, n) = my(sj=Str(k)); for(j=1, n, if (!isprime(eval(sj=concat(sj, Str(3)))), 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) + "3"*k)) for k in range(1, n+1)):
        prefix += 1
    return prefix
print([a(n) for n in range(9)]) # Michael S. Branicky, Nov 18 2021

Crossrefs

Cf. A185682, A185684, A185685, A185687.

Sequence in context: A216668 A057330 A207820 * A280842 A244442 A338700

Adjacent sequences: A186066 A186067 A186068 * A186070 A186071 A186072

Keyword

nonn,base,hard,more

Author

Michel Lagneau, Feb 11 2011

Extensions

a(10)-a(12) from Jonathan Pappas, Oct 19 2021

Status

approved