A241019 -id:A241019 - cp's OEIS frontend

A241018 a(n) is the smallest j such that the n-digit number consisting of a 1 in position j and 9's in the other n-1 positions is a prime, or 0 if no such prime exists.

1, 1, 1, 5, 1, 7, 1, 0, 2, 6, 3, 3, 11, 2, 14, 4, 0, 4, 6, 0, 4, 20, 6, 7, 18, 1, 1, 23, 8, 8, 23, 7, 0, 0, 0, 26, 33, 0, 11, 8, 5, 8, 13, 12, 44, 2, 2, 0, 11, 31, 17, 39, 1, 51, 5, 7, 4, 29, 9, 16, 0, 0, 26, 14, 26, 10, 13, 0, 0, 34, 40, 0, 15, 3, 14, 32, 0

Offset: 2

Michel Lagneau, Apr 15 2014

Previous name: Let x(1)x(2)...x(n) denote the decimal expansion of a number p having an index j such that x(j) = 1 and x(i) = 9 for i <> j. a(n) is the smallest index j such that p is prime, or 0 if no such prime exists.

Except 0, the corresponding primes are 19, 199, 1999, 99991, 199999, 9999991, 19999999, 0, 9199999999, 99999199999, 991999999999, 9919999999999, ... .

Robert Israel, Table of n, a(n) for n = 2..4033

Cf. A241019, A241020.

with(numtheory):nn:=80:T:=array(1..nn):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=9:
     od:
       ii:=0:
       for j from 1 to n while(ii=0)do:
       T[j]:=1:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true
         then
         ii:=1: printf(`%d, `,j):
         else
         T[j]:=9:
         fi:
         od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
       od:

Table[With[{w = ConstantArray[9, n]}, SelectFirst[Range@ n, PrimeQ@ FromDigits@ ReplacePart[w, # -> 1] &]] /. m_ /; MissingQ@ m -> 0, {n, 2, 78}] (* Michael De Vlieger, Sep 13 2017 *)

from sympy import isprime
def a(n):
    base = (10**n-1)
    for j in range(1, n+1):
        t = base - 8*10**(n-j)
        if isprime(t):
            return j
    return 0
print([a(n) for n in range(2, 78)]) # Michael S. Branicky, Jun 02 2024

Name simplified by Jon E. Schoenfield, Sep 13 2017

A241020 a(n) is the smallest j such that the n-digit number consisting of a 1 in position j and 7's in the other n-1 positions is a prime, or 0 if no such prime exists.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 6, 0, 1, 2, 0, 2, 1, 0, 3, 0, 0, 5, 2, 0, 6, 4, 0, 7, 4, 0, 12, 0, 0, 19, 8, 0, 26, 5, 0, 0, 33, 0, 6, 11, 0, 1, 23, 0, 18, 34, 0, 15, 0, 0, 1, 22, 0, 1, 50, 0, 32, 15, 0, 15, 25, 0, 21, 10, 0, 29, 47, 0, 0, 11, 0, 56, 14, 0, 2, 0, 0, 54, 3

Offset: 2

Michel Lagneau, Apr 15 2014

Previous name: Let x(1)x(2)... x(n) denote the decimal expansion of a number p having an index j such that x(j) = 1 and x(i) = 7 for i <> j. The sequence lists the smallest index j such that p is prime, or 0 if no such prime exists.

Except 0, the corresponding primes are 17, 0, 1777, 71777, 0, 0, 77777177, 0, 1777777777, 71777777777, 0, 7177777777777, 17777777777777, 0, 7717777777777777, 0, 0, 7777177777777777777, ... .

Robert Israel, Table of n, a(n) for n = 2..4000

Cf. A241018, A241019.

with(numtheory):nn:=80:T:=array(1..nn):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=7:
     od:
       ii:=0:
       for j from 1 to n while(ii=0)do:
       T[j]:=1:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true
         then
         ii:=1: printf(`%d, `,j):
         else
         T[j]:=7:
         fi:
         od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
       od:

Flatten[Position[IntegerDigits[#],1]&/@Table[Select[FromDigits/@Permutations[ Join[ {1},PadRight[ {},n,7]]],PrimeQ]/.{}->{0,0},{n,80}][[;;,1]]/.{}->0] (* Harvey P. Dale, Jul 21 2024 *)

from sympy import isprime
def a(n):
    if (1+7*(n-1))%3 == 0:
        return 0
    base = (10**n-1)//9*7
    for j in range(1, n+1):
        t = base - 6*10**(n-j)
        if isprime(t):
            return j
    return 0
print([a(n) for n in range(2, 81)]) # Michael S. Branicky, Jun 02 2024

a(n) = 0 when 7*(n-1) + 1 mod 3 = 0. - Michael S. Branicky, Jun 02 2024

Name simplified by Michael S. Branicky, Jun 02 2024

A241022 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 3 for i<>j, or 0 if no such prime exists.

Original entry on oeis.org

13, 313, 3313, 31333, 313333, 3331333, 31333333, 333331333, 3333133333, 33331333333, 333313333333, 0, 33133333333333, 333333313333333, 1333333333333333, 33333333331333333, 333333133333333333, 3333313333333333333, 33313333333333333333, 333313333333333333333

Offset: 2

Michel Lagneau, Apr 15 2014

The corresponding index of the decimal digit 1 are 1, 2, 3, 2, 2, 4, 2, 6, 5, 5, 5, 0, 3, 8, 1, 11, 7, 6, 4, 5,...(A241019).

Michel Lagneau, Table of n, a(n) for n = 2..150

Cf. A241018, A241019, A241020, A241021, A241027.

with(numtheory):nn:=80:T:=array(1..nn):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=3:
     od:
       ii:=0:
       for j from 1 to n while(ii=0)do:
       T[j]:=1:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true
         then
         ii:=1: printf(`%d, `,s):
         else
         T[j]:=3:
         fi:
       od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
     od:

A241021 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 9 for i<>j, or 0 if no such prime exists.

Original entry on oeis.org

19, 199, 1999, 99991, 199999, 9999991, 19999999, 0, 9199999999, 99999199999, 991999999999, 9919999999999, 99999999991999, 919999999999999, 9999999999999199, 99919999999999999, 0, 9991999999999999999, 99999199999999999999, 0, 9991999999999999999999

Offset: 2

Michel Lagneau, Apr 15 2014

The corresponding indices of the decimal digit 1 are 1, 1, 1, 5, 1, 7, 1, 0, 2, 6, 3, 3, 11, 2, 14, 4, 0, 4, 6, 0, 4, ... (A241018).

Michel Lagneau, Table of n, a(n) for n = 2..150

Cf. A241018, A241019, A241020, A241022, A041027.

with(numtheory):nn:=80:T:=array(1..nn):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=9:
     od:
       ii:=0:
       for j from 1 to n while(ii=0)do:
       T[j]:=1:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true
         then
         ii:=1: printf(`%d, `,s):
         else
         T[j]:=9:
         fi:
         od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
     od:

Table[SelectFirst[FromDigits/@Table[Insert[PadRight[{},k,9],1,n],{n,k+1}],PrimeQ],{k,30}]/.Missing["NotFound"]->0 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 10 2017 *)

A241027 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 7 for i<>j, or 0 if no such prime exists.

Original entry on oeis.org

17, 0, 1777, 71777, 0, 0, 77777177, 0, 1777777777, 71777777777, 0, 7177777777777, 17777777777777, 0, 7717777777777777, 0, 0, 7777177777777777777, 71777777777777777777, 0, 7777717777777777777777, 77717777777777777777777, 0, 7777771777777777777777777

Offset: 2

Michel Lagneau, Apr 15 2014

The corresponding index of the decimal digit 1 are 1, 0, 1, 2, 0, 0, 6, 0, 1, 2, 0, 2, 1, 0, 3, 0, 0, 5, 2,...(A241020).

Michel Lagneau, Table of n, a(n) for n = 2..150

Cf. A241018, A241019, A241020, A241022.

with(numtheory):nn:=80:T:=array(1..nn):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=7:
     od:
       ii:=0:
       for j from 1 to n while(ii=0)do:
       T[j]:=1:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true
         then
         ii:=1: printf(`%d, `,s):
         else
         T[j]:=7:
         fi:
       od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
     od:

cp's OEIS Frontend

A241018 a(n) is the smallest j such that the n-digit number consisting of a 1 in position j and 9's in the other n-1 positions is a prime, or 0 if no such prime exists.

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A241020 a(n) is the smallest j such that the n-digit number consisting of a 1 in position j and 7's in the other n-1 positions is a prime, or 0 if no such prime exists.

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A241022 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 3 for i<>j, or 0 if no such prime exists.

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A241021 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 9 for i<>j, or 0 if no such prime exists.

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Maple

Mathematica

A241027 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 7 for i<>j, or 0 if no such prime exists.

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Programs

Maple