A065580 -id:A065580 - cp's OEIS frontend

A037059 Smallest prime containing exactly n 3's.

2, 3, 233, 2333, 23333, 313333, 3233333, 31333333, 333233333, 3233333333, 23333333333, 333313333333, 3333333333383, 33133333333333, 323333333333333, 1333333333333333, 23333333333333333, 333333133333333333, 3333313333333333333, 33313333333333333333

Offset: 0

Patrick De Geest, Jan 04 1999

For almost all n >= 0, a(n) equals [10^(n+1)/3] with one of the (first) digits 3 replaced by a digit 1 or 2. We conjecture that in the few other cases (e.g., for n = 12, 119, ...) the statement holds with some digit 3 replaced by a digit among {4, 5, 7, 8}, except for the special case a(1) = 3. - M. F. Hasler, Feb 22 2016

M. F. Hasler, Table of n, a(n) for n = 0..200; a(1) through a(100) from Harvey P. Dale.

Different from A065580.
Cf. A065586, A065580, A037058, A034388, A036507-A036536.
Cf. A037053, A037055, A037057, A037061, A037063, A037065, A037067, A037069, A037071.

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, p = Permutations[ Join[ Table[ b, {i, 1, n}], {x, y}]]; q = {}; Do[q = Append[q, Replace[p, {x -> c[[i]], y -> c[[j]]}, 2]], {i, 1, 9}, {j, 1, 9}]; Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]]]; Table[ f[n, 3], {n, 1, 18}]
Table[Sort[Flatten[Table[Select[FromDigits/@Permutations[Join[{n},PadRight[{},i,3]]], PrimeQ],{n,0,9}]]][[1]],{i,20}] (* Harvey P. Dale, Feb 28 2015 *)

A037059(n)={if(n==1,3,my(t=10^(n+1)\3); forvec(v=[[-1, n], [-2, -1]], ispseudoprime(p=t+10^(n-v[1])*v[2]) && return(p)); forvec(v=[[0, n], [1, 5]], ispseudoprime(p=t+10^v[1]*v[2]) && return(p)))} \\ M. F. Hasler, Feb 22 2016

a(n) = prime(A037058(n)). - Amiram Eldar, Jul 21 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

More terms and a(0) = 2 prefixed by M. F. Hasler, Feb 22 2016

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

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

Original entry on oeis.org

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

A065581 Smallest prime ending in exactly n 7's.

Original entry on oeis.org

7, 277, 1777, 47777, 2477777, 16777777, 137777777, 577777777, 1777777777, 67777777777, 377777777777, 16777777777777, 17777777777777, 577777777777777, 2777777777777777, 157777777777777777, 377777777777777777, 2777777777777777777, 97777777777777777777, 2477777777777777777777

Offset: 1

Robert G. Wilson v, Nov 28 2001

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

Cf. A037067, A065590, A065821, A065580, A065582.

Do[a = Table[7, {n} ]; k = 0; While[ b = FromDigits[ Join[ IntegerDigits[k], a]]; Mod[k, 10] == 7 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]
k7[n_]:=Module[{c=FromDigits[PadRight[{},n,7]],k=0},While[Nand[PrimeQ[k*10^n + c], Mod[k, 10] != 7],k++];k*10^n+c]; Array[k7,20] (* Harvey P. Dale, Jan 29 2013 *)

a(n)={ my(t=10^n, b=7*(t-1)/9, d=0); while (!isprime(b + t*d), d++; if(d%10==7, d++)); b + t*d } \\ Harry J. Smith, Oct 23 2009

A065586 Smallest prime beginning with exactly n 3's.

Original entry on oeis.org

2, 3, 331, 3331, 33331, 333331, 3333331, 33333331, 3333333319, 33333333329, 333333333323, 3333333333301, 33333333333319, 333333333333307, 3333333333333301, 33333333333333323, 333333333333333391, 333333333333333331, 33333333333333333359, 3333333333333333333041

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037059, A065580, A065584 - A065592.

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A176183 Smallest prime p = p(n) ending with exactly n strings "13".

Original entry on oeis.org

13, 21313, 4131313, 1213131313, 81313131313, 1131313131313, 613131313131313, 21313131313131313, 8131313131313131313, 113131313131313131313, 171313131313131313131313, 23131313131313131313131313

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 11 2010

See references of A176096.

List of prefixes: for 1st term p(1) = 13: ./., 2, 4, 12, 8, 1, 6, 2, 8, 1, 17, 23, 21, 8, 59, 87, 53, 46, 10, 8, 20, 73, 29, 20, 3, 2, 16, 33, 80, 2, ...

			n = 1: prime(6) = 13 is 1st term of sequence.
n = 2: prime(2392) = 21313 is 2nd term of sequence.
Note: 1st such palindromic prime (i.e. prefix 3): p(25) = 3(13)_25 = 313131313131313131313131313131313131313131313131313.

Cf. A065580, A065585, A176096, A176119.

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A037059 Smallest prime containing exactly n 3's.

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A065582 Smallest prime ending in exactly n 9's.

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

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A065581 Smallest prime ending in exactly n 7's.

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A065586 Smallest prime beginning with exactly n 3's.

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A176183 Smallest prime p = p(n) ending with exactly n strings "13".

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