A187614 -id:A187614 - cp's OEIS frontend

A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

11, 13, 17, 19, 21, 23, 29, 31, 34, 38, 41, 42, 43, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 76, 77, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 94, 95, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118

Offset: 1

Andrew Slattery, Mar 12 2020

Almost every prime appears in this sequence.
Among the first 10000 primes, only 2, 3, 5, 7, 37, 79, 239, 4649, and 62003 do not appear in the sequence. - Giovanni Resta, Mar 13 2020
The next primes not in the sequence are 538987, 35121409, and 265371653. - Robert Israel, Mar 18 2020

			5 is not in the sequence because 1/5 = 0.2 does not contain any 9s.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Subsequences (for terms > 1): A000533, A002275, A135577, A252491.
Cf. A216664 (a subsequence).
Cf. A187614.

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:
select(f, [$1..1000]); # Robert Israel, Mar 18 2020

Select[Range[120], MemberQ[ Flatten@ RealDigits[1/#][[1]], 9] &] (* Giovanni Resta, Mar 12 2020 *)

from itertools import count, islice
from sympy import n_order, multiplicity
def A333237_gen(startvalue=1): # generator of terms
    for m in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,m), multiplicity(5,m)
        if max(str(10**(max(m2,m5)+n_order(10,m//2**m2//5**m5))//m)) == '9':
            yield m
A333237_list = list(islice(A333237_gen(), 10)) # Chai Wah Wu, Feb 07 2022

A333236(a(n)) = 9.

More terms from Giovanni Resta, Mar 12 2020

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

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Feb 28 2022

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

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

Subsequence of A187614.

Cf. A004023, A333237, A352024.

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

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

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

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

A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

Original entry on oeis.org

9091, 909091, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091, 909090909090909090909090909090909090909090909090909090909090909091

Offset: 1

Rick L. Shepherd, Jul 30 2004

nonn

Equivalently, primes of the form 9090...9091 (A054416(n)-1 copies of 90 followed by 91), the subsequence of all primes in A095372.

These primes appear in A187614 because the decimal representation of their reciprocal contains only the digits 0, 1, 8, and 9.

Cf. A001562, A054416, A095372.

A187372 Numbers k such that the decimal digits of 1/k contain every digit at least once.

Original entry on oeis.org

17, 19, 23, 29, 34, 38, 46, 47, 49, 51, 53, 57, 58, 59, 61, 68, 69, 71, 76, 83, 85, 87, 89, 92, 94, 95, 97, 98, 102, 103, 107, 109, 113, 114, 115, 116, 118, 119, 121, 122, 127, 129, 131, 133, 136, 138, 139, 141, 142, 145, 147, 149, 151, 152, 153, 157, 161, 163, 166, 167, 169, 170, 171, 173, 174, 177, 178, 179, 181, 183, 184, 188

Offset: 1

Michel Lagneau, Mar 09 2011

			17 is in the sequence because 1/17 = .0588235294117647 0588235294117647 ...
contains every digit at least once ;
31 is not in the sequence because 1/31 = .032258064516129 032258064516129...
without the digit 7.

Cf. A187614.

with(numtheory):Digits:=200:B:={0, 1, 2, 3, 4, 5, 6, 7, 8, 9}: T:=array(1..250)
  : for p from 1 to 200 do:ind:=0:n:=floor(evalf(10^200/p)):l:=length(n):n0:=n:s:=0:for
  m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v : T[m]:=u:od: A:=convert(T,
  set):z:=nops(A):if A intersect B = B and ind=0 then ind:=1: printf(`%d, `, p):else
  fi:od:

A2 := {}; Do[ If[Length[Union[IntegerDigits[Floor[10^200/n]]]] == 10, A2 =
  Join[A2, {n}]], {n, 1, 200}]; Print[A2]

A187868 Primes of the form 100^k - 10^k + 1.

Original entry on oeis.org

9901, 99990001, 999999000001, 9999999900000001

Offset: 1

T. D. Noe, Mar 14 2011

These primes correspond to k = 2, 4, 6, and 8. There are no other primes of this form for k up to 10000. These primes appear in A187614 because the decimal representation of their reciprocal contains only the digits 0, 1, 8, and 9.
Because these numbers are values of the sixth cyclotomic polynomial, by Theorem 1 of Golomb, the only k that can produce primes are the 3-smooth numbers, 2^i*3^j (A003586).
Next term has k > 1000000. - Robert Gelhar, Aug 20 2020

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 85.

Solomon W. Golomb, Cyclotomic polynomials and factorization theorems, Amer. Math. Monthly 85 (1978), 734-737.

Cf. A003586, A168624, A187614.

Select[Table[100^n - 10^n + 1, {n, 1000}], PrimeQ]

A352024 Largest digit in the decimal expansion of 1/A352023(n).

Original entry on oeis.org

5, 3, 2, 8, 7, 8, 8, 5, 8, 8, 8, 8

Offset: 1

Bernard Schott, Mar 01 2022

All terms are < 9.

A352023(13) <= 5363222357 and A352023(14) <= 77843839397, in both cases, the corresponding largest digit in the decimal expansion of the inverse is 8.

			A352023(5) = 37, the largest digit in the decimal expansion of 1/37 = 0.027027027027027... is 7, hence a(5) = 7.

Cf. A187614, A333236, A333237, A352023.

a(n) = A333236(A352023(n)). - Amiram Eldar, Mar 02 2022

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A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

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

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A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

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A187372 Numbers k such that the decimal digits of 1/k contain every digit at least once.

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A187868 Primes of the form 100^k - 10^k + 1.

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A352024 Largest digit in the decimal expansion of 1/A352023(n).

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