cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A037067 Smallest prime containing exactly n 7's.

Original entry on oeis.org

2, 7, 277, 1777, 47777, 727777, 7477777, 77767777, 577777777, 1777777777, 67777777777, 377777777777, 7177777777777, 17777777777777, 577777777777777, 2777777777777777, 77777767777777777, 377777777777777777, 2777777777777777777, 71777777777777777777
Offset: 0

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Comments

We conjecture that for all n >= 2, a(n) equals floor(10^(n+1)/9)*7 with one of the (first) digits 7 replaced by a digit among {0, ..., 6}. - M. F. Hasler, Feb 22 2016
The conjecture is false: a(668) = 7*(10^669-1)/9 + 10^276. - Robert Israel, Jul 13 2016

Links

Crossrefs

Cf. A065590, A065581, A037066, A034388, A036507-A036536.
Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037069, A037071.

Programs

  • Maple
    F:= proc(n) local x0,i,j;
  x0:= 7/9*(10^(n+1)-1);
  for j from 1 to 6 do
    if isprime(x0 + (j-7)*10^n) then
      return x0 + (j-7)*10^n fi od;
  for i from n-1 to 0 by -1 do
    for j from 0 to 6 do
     if isprime(x0 + (j-7)*10^i) then
       return x0 + (j-7)*10^i fi od od;
  for i from 0 to n do
    for j from 8 to 9 do
       if isprime(x0 + (j-7)*10^i) then
         return x0 + (j-7)*10^i fi
  od od:
end proc:
F(0):= 2: F(1):= 7:
map(F, [$0..100]); # Robert Israel, Jul 13 2016
  • Mathematica
    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, 7], {n, 1, 18}]
  • PARI
    A037067(n)={my(t=10^(n+1)\9*7); forvec(v=[[-1, n], [-7, -1]], ispseudoprime(p=t+10^(n-v[1])*v[2]) && return(p)); error} \\ M. F. Hasler, Feb 22 2016

Formula

a(n) = prime(A037066(n)). - Amiram Eldar, Jul 21 2025

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003
More terms from and a(0) = 2 prepended 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

Views

Author

Robert G. Wilson v, Nov 28 2001

Keywords

Comments

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)

Links

Crossrefs

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

Programs

  • Mathematica
    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} ]
  • PARI
    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
  • Python
    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

Views

Author

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

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)_
  • PARI
    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

Extensions

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

A065580 Smallest prime ending in exactly n 3's.

Original entry on oeis.org

3, 233, 2333, 23333, 733333, 10333333, 83333333, 1033333333, 17333333333, 23333333333, 2633333333333, 10333333333333, 53333333333333, 3233333333333333, 1333333333333333, 23333333333333333, 2033333333333333333, 10333333333333333333, 173333333333333333333, 1733333333333333333333
Offset: 1

Views

Author

Robert G. Wilson v, Nov 28 2001

Keywords

Links

Crossrefs

Cf. A037059, A065586, A065821, A065581, A065582.

Programs

  • Mathematica
    Do[a = Table[3, {n} ]; k = 0; While[ b = FromDigits[ Join[ IntegerDigits[k], a]]; Mod[k, 10] == 3 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]
  • PARI
    a(n)={ my(t=10^n, b=(t-1)/3, d=0); while (!isprime(b + t*d), d++; if(d%10==3, d++)); b + t*d } \\ Harry J. Smith, Oct 23 2009

A065590 Smallest prime beginning with exactly n 7's.

Original entry on oeis.org

2, 7, 773, 77711, 77773, 7777709, 77777719, 777777701, 777777773, 77777777717, 777777777713, 777777777773, 7777777777771, 777777777777719, 777777777777773, 77777777777777711, 777777777777777737, 7777777777777777793, 77777777777777777729, 77777777777777777771
Offset: 0

Views

Author

Robert G. Wilson v, Nov 28 2001

Keywords

Links

Crossrefs

Cf. A037067, A065581, A065584 - A065592.

Extensions

Corrected by Don Reble, Jan 17 2007
Offset corrected by Sean A. Irvine, Sep 06 2023
Showing 1-5 of 5 results.

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