A037055 -id:A037055 - cp's OEIS frontend

A065821 a(n) is the smallest prime ending in exactly n 1's.

31, 11, 2111, 101111, 311111, 29111111, 61111111, 1711111111, 14111111111, 31111111111, 311111111111, 2111111111111, 31111111111111, 3511111111111111, 5111111111111111, 101111111111111111, 3511111111111111111, 2111111111111111111, 1111111111111111111, 911111111111111111111

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Nov 23 2001

			a(4) = 101111 because 1111=11*101, 21111=3*31*227, 31111=53*587, 41111=7^2*829, 51111=3^4*631, 61111=23*2657, 71111=17*47*89, 81111=3*19*1423, 91111=179*509 so 101111 is the first prime ending in four 1's.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A037055, A065584, A065580, A065581, A065582.

pe[n_]:=Module[{k=0,len=IntegerLength[n]},While[Mod[k,10]==1||(!PrimeQ[ k*10^len+n]),k++];k*10^len+n]; pe/@Table[(10^n-1)/9,{n,20}] (* Harvey P. Dale, Dec 31 2013 *)_

a(n)={ my(f=10^n, b=(f-1)/9, k=0); while (!isprime(b + k*f), k+=1+(k%10==0)); b + k*f } \\ Harry J. Smith, Nov 01 2009

Edited and extended by Robert G. Wilson v, Jul 04 2003

A037054 a(n)-th prime is the smallest prime containing exactly n 1's.

Original entry on oeis.org

1, 6, 5, 187, 1242, 9682, 86538, 733339, 5821735, 56196114, 503193257, 4161915701, 41621368333, 383118399789, 3549047966306, 33056584174792, 309353882119895, 2651938403956789, 27417323062119921, 27417323062119920, 2461813897281353902, 23422580231698331842

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, A037055, A034388.

Cf. A037052, A037056, A037058, A037060, A037062, A037064, A037066, A037068, A037070.

(* see A037055 for f *) PrimePi[ Table[ f[n, 1], {n, 1, 13}]]

a(n) = A000720(A037055(n)). - Amiram Eldar, Jul 20 2025

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 20 2025

A084673 Smallest prime in which a digit appears n times.

Original entry on oeis.org

2, 11, 1117, 10111, 101111, 1111151, 11110111, 101111111, 1111111121, 11111111113, 101111111111, 1111111118111, 11111111111411, 111111111116111, 1111111111111181, 11111111101111111, 101111111111111111

Offset: 1

Harvey P. Dale, Jun 29 2003

For n > 1, conjectured to be equal to A037055(n), the smallest prime in { R-10^n, R-10^(n-1), ..., R-10; R+a*10^b, a = 1, ..., 8, b = 0, 1, 2, ..., n }, where R = (10^(n+1)-1)/9 is the (n+1)-digit repunit. - M. F. Hasler, Feb 25 2016

			a(4)=10111 because 10111 is the smallest prime with four duplicate digits.

Liz Strachan, Numbers are Forever, Mathematical Facts and Curiosities, Constable, London, 2014, page 267.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A034388, A037055, A065584, A065821.

Table[ First[ Select[ Prime[ Range[100000]], Max[ DigitCount[ # ]]==i & ]], {i, 6}] (* or *)
f[n_, b_] := Block[{k = 10^(n + 1), p = Permutations[ Join[ Table[ b, {i, 1, n}], {x}]], c = Complement[ Table[j, {j, 0, 9}], {b}], q = {}}, Do[q = Append[q, Replace[p, x -> c[[i]], 2]], {i, 1, 9}]; r = Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]; If[ r != Infinity, r, While[ !PrimeQ[k] || Count[ IntegerDigits[k], b] != n, k++ ]; k]]; Table[ f[n, 1], {n, 2, 18}]

A084673(n)=if(n>1,A037055(n),2) \\ M. F. Hasler, Feb 25 2016

Edited and extended by Robert G. Wilson v, Jul 03 2003

A176096 Smallest prime p = p(n) containing exactly n strings "13" (n = 1, 2, ...).

Original entry on oeis.org

13, 13313, 1313813, 131313113, 13131313133, 1131313131313, 131313131313139, 13131313131313913, 1313131311313131313, 113131313131313131313, 13131313131313133131313, 1313131131313131313131313

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 08 2010

			n = 1: prime(6) = 13 is 1st term of sequence
prime(12268) = 131303 > 21313 = prime(2392) > 13313 = prime(1581) = p(2), 2nd term of sequence
prime(857198) = 13131317 > 4131313 = prime(291796) > prime(102949) = 1341313 > 1313813 = prime() = p(3), 3rd term of sequence
n = 13: 131131313131313131313131313 a 27-digit prime is 13th term of sequence

E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig/Jena/Berlin 1982
B. A. Kordemski: Koepfchen, Koepfchen! Mathematik zur Unterhaltung, Urania Verlag Leipzig/Jena/Berlin 1965

A037053, A037055, A037057, A037059, A037063, A037067

A177999 Largest n digit prime with the most digits equal to 1.

Original entry on oeis.org

7, 11, 911, 8111, 16111, 911111, 1171111, 71111111, 131111111, 1711111111, 31111111111, 311111111111, 5111111111111, 41111111111111, 111151111111111, 5111111111111111, 11111611111111111, 191111111111111111

Offset: 1

Lekraj Beedassy, May 17 2010

Select first for maximum number of 1's, then take the largest.

In more detail: To get a(n), look at the list of all the n-digit primes. Suppose k is the maximum number of 1's of any number on the list. Throw out any prime on the list that does not contain k 1's. Then a(n) = maximal number that is left on the list. - N. J. A. Sloane, Mar 20 2018

Cf. A004022, A037055, A105992, A178000.

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 *)

cp's OEIS Frontend

A065821 a(n) is the smallest prime ending in exactly n 1's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A037054 a(n)-th prime is the smallest prime containing exactly n 1's.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A084673 Smallest prime in which a digit appears n times.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A176096 Smallest prime p = p(n) containing exactly n strings "13" (n = 1, 2, ...).

Views

Author

Keywords

Examples

References

Crossrefs

A177999 Largest n digit prime with the most digits equal to 1.

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica