A352023 Primes p such that 1/p does not contain digit '9' in its decimal expansion.

2, 3, 5, 7, 37, 79, 239, 4649, 62003, 538987, 35121409, 265371653

Offset: 1

Author

Bernard Schott, Feb 28 2022

Keywords

• nonn
• base
• more

Comments

Terms a(1)-a(9) and a(10)-a(12) were respectively found by Giovanni Resta and Robert Israel (comments in A333237).

The corresponding largest digit in the decimal expansion of 1/a(n) is A352024(n).

If it exists, a(13) > 2.7*10^8.

a(13) > 1360682471 (with A187614). - Jinyuan Wang, Mar 03 2022

a(13) <= 5363222357, a(14) <= 77843839397. - David A. Corneth, Mar 03 2022

Examples

			The largest digit in the decimal expansion of 1/7 = 0.142857142857... is 8 < 9, hence 7 is a term.
		

Crossrefs

Subsequence of A187614.

Cf. A004023, A333237, A352024.

Programs

Maple

f:= proc(n) local m, S, r;
   m:= 1; S:= {1};
   do
     r:= floor(m/n);
     if r = 9 then return true fi;
     m:= (m - r*n)*10;
     if member(m, S) then return false fi;
     S:= S union {m};
   od
end proc:
remove(f, [seq(ithprime(i),i=1..10^5)]); # Robert Israel, Mar 16 2022

Mathematica

Select[Range[10^5], PrimeQ[#] && FreeQ[RealDigits[1/#][[1, 1]], 9] &] (* Amiram Eldar, Feb 28 2022 *)

PARI

isok(p) = if (isprime(p), my(m2=valuation(p, 2), m5=valuation(p, 5)); vecmax(digits(floor(10^(max(m2,m5) + znorder(Mod(10, p/2^m2/5^m5))+1)/p))) < 9); \\ Michel Marcus, Feb 28 2022

Python

from sympy import n_order, nextprime
from itertools import islice
def A352023_gen(): # generator of terms
    yield from (2,3,5)
    p = 7
    while True:
        if '9' not in str(10**(n_order(10, p))//p):
            yield p
        p = nextprime(p)
A352023_list = list(islice(A352023_gen(),9)) # Chai Wah Wu, Mar 03 2022