A037071 -id:A037071 - cp's OEIS frontend

A065592 Smallest prime beginning with exactly n 9's.

2, 97, 991, 99901, 99991, 9999901, 9999991, 999999929, 9999999929, 99999999907, 999999999937, 9999999999971, 99999999999923, 999999999999947, 9999999999999917, 99999999999999919, 99999999999999997, 9999999999999999919, 99999999999999999931

Offset: 0

Robert G. Wilson v, Nov 28 2001

M. F. Hasler, Table of n, a(n) for n = 0..200

Cf. A037071, A065582, A065584 - A065591.

fp[n_]:=Select[Join[10*n+{1,7},100*n+Range[1,99,2]],PrimeQ,1]; With[{ns=Table[FromDigits[PadRight[{},n,9]],{n,20}]}, Join[{2}, Flatten[fp/@ns]]] (* Harvey P. Dale, May 12 2012 *)

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A065582 Smallest prime ending in exactly n 9's.

Original entry on oeis.org

19, 199, 1999, 49999, 199999, 2999999, 19999999, 799999999, 10999999999, 59999999999, 1099999999999, 34999999999999, 59999999999999, 499999999999999, 14999999999999999, 139999999999999999, 1099999999999999999, 20999999999999999999, 29999999999999999999, 2099999999999999999999

Offset: 1

Robert G. Wilson v, Nov 28 2001

From Jeppe Stig Nielsen, Jul 30 2022: (Start)
Can decrease, for example a(25) < a(24). So not the same as Smallest prime ending in n or more 9s.
a(n) can contain other 9s as well, for example a(46), a(118), a(156). (End)

Hugo Pfoertner, Table of n, a(n) for n = 1..500 (first 100 terms from Harry J. Smith).

Cf. A037071, A065592, A065821, A065580, A065581.

Do[a = Table[9, {n} ]; k = 0; While[ b = FromDigits[ Join[ IntegerDigits[k], a]]; Mod[k, 10] == 9 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]

a(n)={ my(t=10^n, b=t-1, d=0); while(!isprime(b + t*d), d++; if(d%10==9, d++)); b + t*d } \\ Harry J. Smith, Oct 23 2009

from sympy import isprime
def a(n):
    pow, end, i = 10**n, 10**n-1, 1
    while i%10 == 9 or not isprime(i*pow + end): i += 1
    return i*pow + end
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jul 30 2022

A037070 a(n)-th prime is the smallest prime containing exactly n 9's.

Original entry on oeis.org

1, 8, 46, 303, 5133, 17984, 216816, 1270607, 41146179, 420243162, 2524038155, 36159205628, 343392568900, 1955010428258, 15237833654620, 260219446617109, 2621513397605657, 24619309639366177, 233874804775621799, 684559920583084690, 20920441130654929928, 200085344903558463823

Offset: 0

Patrick De Geest, Jan 04 1999

Andrew R. Booker, The Nth Prime Page.
Kim Walisch, primecount, Fast prime counting function library.

Cf. A000720, A037071, A034388.

Cf. A037052, A037054, A037056, A037058, A037060, A037062, A037064, A037066, A037068.

(* see A037071 for f *) PrimePi[ Table[ f[n, 9], {n, 1, 13}]]

a(n) = A000720(A037071(n)). - Amiram Eldar, Jul 21 2025

One more term from Vladeta Jovovic, Jan 10 2002
a(0)=1 prepended by Sean A. Irvine, Dec 06 2020
a(14)-a(21) calculated using Kim Walisch's primecount and added by Amiram Eldar, Jul 21 2025

A268707 Smallest n-digit prime having at least n-1 digits equal to 9.

Original entry on oeis.org

2, 19, 199, 1999, 49999, 199999, 2999999, 19999999, 799999999, 9199999999, 59999999999, 959999999999, 9919999999999, 59999999999999, 499999999999999, 9299999999999999, 99919999999999999, 994999999999999999, 9991999999999999999, 29999999999999999999

Offset: 1

Michel Lagneau, Michael De Vlieger and Robert G. Wilson v, Feb 11 2016

Michel Lagneau, Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1225

Cf. A037071, A241100, A268702 - A268706, A241206, A266148.

f[n_] := Block[{k = 0, p = {}, r = (10^n - 1), s = Range@ 10 - 10}, While[k < n - 0, AppendTo[p, Select[r + 10^k*s, PrimeQ]]; k++]; p = Min@ Flatten@ p]; Array[f, 20]

a(n)=my(t=10^n-1,p); forstep(d=n-1,0,-1, forstep(k=8,1,-1, p=t-10^d*k; if(ispseudoprime(p), return(p)))); -1 \\ Charles R Greathouse IV, Mar 21 2016

A178007 Largest n-digit prime with the most digits equal to 9.

Original entry on oeis.org

7, 97, 997, 9949, 99991, 999979, 9999991, 99999989, 999999929, 9999999929, 99999999599, 999999999989, 9999999999799, 99999999999959, 999999999999989, 9999999999999199, 99999999999999997, 999999999999999989, 9999999999999999919, 99999999999999999989, 999999999999999999899, 9999999999999999999929

Offset: 1

Lekraj Beedassy, May 17 2010

First maximum the number of 9's, then choose the largest.
From Robert Israel, Dec 18 2024: (Start)
This is believed to be different from A241206, as there should be infinitely many n for which there is no n-digit prime with n-1 digits equal to 9.  No examples are known; the least such n is greater than 3400. (End)

Robert Israel, Table of n, a(n) for n = 1..996

Cf. A037071, A105975, A003618, A241206, A266148.

f:= proc(n) local i,j,a,b,x,y;
     x:= 10^n-1;
     for i from 0 to n-1 do
       for a from 1 to 9 do
         y:= x - a*10^i;
         if isprime(y) then return y fi;
     od od;
     for i from 1 to n-1 do
       for a from 1 to 9 do
         for j from 0 to i-1 do
           for b from 1 to 9 do
             y:= x - a*10^i - b*10^j;
             if isprime(y) then return y fi
    od od od od;
    FAIL
end proc:
map(f, [$1..30]); # Robert Israel, Dec 16 2024

Corrected and more terms by Robert Israel, Dec 16 2024

A375760 Array read by rows: T(n,k) is the first prime with exactly n occurrences of decimal digit k.

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 101, 13, 2, 3, 41, 5, 61, 7, 83, 19, 1009, 11, 223, 233, 443, 557, 661, 277, 881, 199, 10007, 1117, 2221, 2333, 4441, 5557, 6661, 1777, 8887, 1999, 100003, 10111, 22229, 23333, 44449, 155557, 166667, 47777, 88883, 49999, 1000003, 101111, 1222229, 313333, 444443, 555557, 666667, 727777, 888887, 199999

Offset: 0

Robert Israel, Aug 27 2024

			T(4,1) = 10111 because 10111 is the first prime with four 1's.
Array starts
      2      2       3      2      2      2      2      2      2      2
    101     13       2      3     41      5     61      7     83     19
   1009     11     223    233    443    557    661    277    881    199
  10007   1117    2221   2333   4441   5557   6661   1777   8887   1999
 100003  10111   22229  23333  44449 155557 166667  47777  88883  49999
1000003 101111 1222229 313333 444443 555557 666667 727777 888887 199999

Robert Israel, Table of n, a(n) for n = 0..1009 (rows 0 to 100)

Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037067, A037069, A037071

F:= proc(v,x) local d,y,z,L,S,SS,Cands,t,i,k;
   for d from v do
     Cands:= NULL;
     if x = 0 then SS:= combinat:-choose([$2..d-1],v)
     elif member(x,[1,3,7,9]) then SS:= combinat:-choose(d,v)
     else SS:= combinat:-choose([$2..d],v)
     fi;
     for S in SS do
       for y from 9^(d-v+1) to 9^(d-v+1)+9^(d-v)-1 do
         L:= convert(y,base,9)[1..d-v+1];
         L:= map(proc(s) if s < x then s else s+1 fi end proc, L);
         i:= 1;
         t:= 0:
         for k from 1 to d do
           if member(k,S) then t:= t + x*10^(k-1)
           else t:= t + L[i]*10^(k-1); i:= i+1;
           fi;
         od;
         Cands:= Cands, t
     od od;
     Cands:= sort([Cands]);
     for t in Cands do if isprime(t) then return t fi od;
   od
end proc:
F(0,0):= 2: F(1,2):= 2: F(1,5):= 5:
for i from 0 to 10 do
  seq(F(i,x), x=0..9)
od;

T[n_,k_]:=Module[{p=2},While[Count[IntegerDigits[p],k]!=n, p=NextPrime[p]]; p]; Table[T[n,k],{n,0,5},{k,0,9}]//Flatten (* Stefano Spezia, Aug 27 2024 *)

A120642 Smallest integer k>0 such that k*10^n - 1 is a prime.

Original entry on oeis.org

2, 2, 2, 5, 2, 3, 2, 8, 11, 6, 11, 35, 6, 5, 15, 14, 11, 21, 3, 21, 14, 6, 6, 80, 8, 2, 2, 6, 9, 48, 48, 21, 15, 6, 44, 11, 9, 15, 18, 6, 33, 30, 3, 278, 74, 92, 89, 33, 8, 71, 59, 11, 2, 5, 3, 24, 108, 47, 39, 41, 6, 14, 53, 173, 26, 26, 51, 114, 23, 17, 246, 44, 6, 131, 56, 8, 26, 77, 74

Offset: 1

Jonathan Vos Post, Aug 17 2006

			The primes are 19, 199, 1999, 49999, 199999, 2999999,
19999999, 799999999, 10999999999, 59999999999, ...,.

Pierre CAMI, Table of n,a(n) for n=1,...,3000

Cf. A030430, A037071, A062800, A065582, A121172.

f[n_] := Block[{k = 1}, While[ !PrimeQ[k*10^n - 1], k++ ]; k]; Array[f, 79] (* Robert G. Wilson v *)

a(11) onwards from Robert G. Wilson v, Aug 20 2006

A120729 Smallest integer k>0 such that k*10^n + 1 is a semiprime.

Original entry on oeis.org

3, 2, 2, 5, 1, 1, 1, 1, 1, 2, 4, 2, 3, 7, 4, 3, 6, 6, 4, 1, 2, 4, 13, 2, 4, 3, 7, 21, 6, 9, 3, 1, 5, 4, 16, 19, 28, 19, 9, 3

Offset: 0

Jonathan Vos Post, Aug 17 2006

The corresponding semiprimes are 4, 21, 201, 5001, 10001, 100001, 100001, 10000001, 2000000001, 40000000001, ... Semiprime analog of A121172.

			a(0) = 3 because 3*10^0 + 1 = 4 = 2^2 is a semiprime.
a(1) = 2 because 2*10^1 + 1 = 21 = 3*7 is a semiprime.
a(2) = 2 because 2*10^2 + 1 = 201 = 3*67 is a semiprime.
a(3) = 5 because 5*10^3 + 1 = 5001 = 3*1667 is a semiprime.
a(4) = 1 because 1*10^4 + 1 = 10001 = 73*137 is a semiprime.
a(5) = 1 because 1*10^5 + 1 = 100001 = 11*9091 is a semiprime.

Cf. A001358, A030430, A037071, A062800, A065582, A121172.

sik[n_]:=Module[{k=1,c=10^n},While[PrimeOmega[k*c+1]!=2,k++];k]; Array[sik,40,0] (* Harvey P. Dale, Aug 20 2012 *)

Smallest integer k>0 such that k*10^n + 1 is in A001358.

More terms from Harvey P. Dale, Aug 20 2012

cp's OEIS Frontend

A065592 Smallest prime beginning with exactly n 9's.

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A065582 Smallest prime ending in exactly n 9's.

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A037070 a(n)-th prime is the smallest prime containing exactly n 9's.

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A268707 Smallest n-digit prime having at least n-1 digits equal to 9.

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PARI

A178007 Largest n-digit prime with the most digits equal to 9.

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Maple

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A375760 Array read by rows: T(n,k) is the first prime with exactly n occurrences of decimal digit k.

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Maple

Mathematica

A120642 Smallest integer k>0 such that k*10^n - 1 is a prime.

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Mathematica

Extensions

A120729 Smallest integer k>0 such that k*10^n + 1 is a semiprime.

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