A241018 -id:A241018 - cp's OEIS frontend

A241019 a(n) is the smallest j such that the n-digit number consisting of a 1 in position j and 3's in the other n-1 positions is a prime, or 0 if no such prime exists.

1, 2, 3, 2, 2, 4, 2, 6, 5, 5, 5, 0, 3, 8, 1, 11, 7, 6, 4, 5, 11, 5, 0, 0, 9, 11, 0, 5, 5, 0, 4, 5, 17, 19, 19, 6, 0, 3, 9, 35, 1, 27, 24, 32, 0, 36, 14, 11, 34, 14, 22, 0, 17, 53, 0, 47, 11, 0, 16, 3, 0, 15, 0, 39, 22, 40, 27, 39, 0, 19, 2, 19, 48, 2, 43, 19

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) = 3 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 13, 313, 3313, 31333, 313333, 3331333, 31333333, 333331333, 3333133333, 33331333333, 333313333333, 0, 33133333333333, ... .

Robert Israel, Table of n, a(n) for n = 2..4001

Cf. A241018, 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]:=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, `,j):
         else
         T[j]:=3:
         fi:
         od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
       od:

from sympy import isprime
def a(n):
    base = (10**n-1)//9*3
    for j in range(1, n+1):
        t = base - 2*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 and offset corrected by Michael S. Branicky, Jun 02 2024

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:

A373351 Numbers k such that there are no primes among the k-digit numbers consisting of one 1 and all other digits 9.

Original entry on oeis.org

1, 9, 18, 21, 34, 35, 36, 39, 49, 62, 63, 69, 70, 73, 78, 80, 86, 93, 99, 102, 116, 122, 123, 141, 142, 143, 146, 147, 150, 178, 182, 185, 201, 206, 223, 230, 236, 238, 241, 246, 251, 267, 279, 285, 292, 293, 304, 309, 313, 321, 326, 329, 343, 346, 347, 350, 355, 362, 375, 380, 381, 385, 389, 398

Offset: 1

Robert Israel, Jun 01 2024

Besides 1, numbers k such that A241018(k) = 0.

			a(2) = 9 is a term because none of the numbers 199999999, 9199999999, ..., 999999991 are prime.

Robert Israel, Table of n, a(n) for n = 1..577

Cf. A241018.

filter:= proc(n) local a,j;
a:= 10^n-1;
not ormap(j -> isprime(a-8*10^j), [$0..n-1])
end proc:
select(filter, [$1..400]);

cp's OEIS Frontend

A241019 a(n) is the smallest j such that the n-digit number consisting of a 1 in position j and 3'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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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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A373351 Numbers k such that there are no primes among the k-digit numbers consisting of one 1 and all other digits 9.

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