A247341 -id:A247341 - cp's OEIS frontend

A232210 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.

1, 0, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 2, 6, 2, 2, 1, 1, 2, 1, 4, 4, 23, 1, 2, 1, 6, 2, 2, 5, 1, 10, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 4, 2, 1, 1, 1, 2, 4, 1, 2, 5, 4, 2, 3, 1, 1, 5, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 4, 2, 14, 2, 4, 1, 3

Offset: 1

Vladimir Shevelev, Sep 13 2014

Conjecture: for n>=3, a(n)>0.
Records are 1,14,23,50,252,4752,...
The corresponding primes are 2,13,131,653,883,1279,...
These primes beginning with the second one we call "stubborn primes".
Counter-conjecture: a(2889)=0. - Hans Havermann, Oct 15 2014
If a(n)=1, then the resulting primes are in A092993 and form A055782; if a(n)=2, then they form sequence 4133,4733,5333,7933,..., etc. - Vladimir Shevelev, Oct 16 2014
If a prime p divides Pb_k, then it also divides Pb_{k+m(p-1)} for all m>=0. This follows from Fermat's little theorem applied to b_x=(10^x-1)/3 with x=p-1. - M. F. Hasler, Oct 20 2014

			For n=1, start with prime(1)=2 and get already at the first step the prime 23. So a(1)=1.
For n=2, starting with prime(2)=3, one never gets a prime by appending further digits "3", therefore a(2)=0.
For n=3, n=4, n=5, one gets after the first step the primes 53, 73, 113, and therefore a(n)=1.
For n=6, start with prime(6)=13; one has to append 14 "3"s in order to get a new prime, so a(6)=14.
For n=2889, start with prime(2889) = 26293. (Do not mix up with prime(2899) = 26393...!) Appending 2k-1 or 6k-4 or 6k-2 or 18k-6 or 36k-18 or 180k-144 digits "3" yields a number divisible by 11 resp. 7 resp. 13 resp. 19 resp. 101 resp. 31. For 18k-12 and 36k (with k <> 1 (mod 5)) digits "3" there is no simple pattern and both yield sometimes large primes in the factorization, but (so far) always composite numbers 26293...3 (up to several thousand digits). - _M. F. Hasler_, Oct 16 2014

Hans Havermann, Table of n, a(n) for n = 1..2888 (first 200 terms from Michel Marcus)
Hans Havermann, "Google+ discussion relating to this sequence"
Vladimir Shevelev, "Stubborn primes"
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found

Cf. A055782, A092993, A242775, A247341, A247342, A248919.

f[n_] := Block[{k = 1, p = Prime@ n}, While[ !PrimeQ[p*10^k + (10^k - 1)/3], k++]; k]; f[2] = 0; Array[f, 100] (* Robert G. Wilson v, Apr 24 2015 *)
m3[n_]:=Module[{k=10n+3},While[!PrimeQ[k],k=10k+3];IntegerLength[k]-IntegerLength[ n]]; Join[{1,0},m3/@Prime[Range[3,90]]] (* Harvey P. Dale, Feb 11 2018 *)

a(n) = {if (n==2, return (0)); p = prime(n); k = 1; while (! isprime(p = p*10+3), k++); k;} \\ Michel Marcus, Sep 13 2014

More terms from Peter J. C. Moses, Sep 13 2014

A242775 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation b_k and prime(n) is prime, or a(n)=0 if there is no such prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 4, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 3, 2, 1, 2, 7, 3, 1, 3, 2, 2, 8, 1, 1, 7, 2, 1, 1, 5, 3, 2, 2, 2, 3, 1, 3, 8, 5, 1, 1, 4, 3, 1, 4, 5, 3, 6, 1, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 6, 3, 1, 3, 4, 2, 3, 8, 4, 1, 3, 34, 1

Offset: 1

Vladimir Shevelev, Sep 13 2014

nonn

Conjecture: for n>=4, a(n)>0.

Records >=1: 1,2,4,7,8,34,... correspond to primes 7,19,41,127,157,443,...

			For n<=3, a(n) = 0, because 3..32, 3..33 and 3..35 can never be prime, whatever the number of 3's that are concatenated.
For n=4, prime(n)=7, 37 is prime. So a(4)=1.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A232210, A247341, A247342.

a(n) = {if (n<=3, return (0)); p = prime(n); k = 1; while (! isprime(p = eval(concat("3", Str(p)))), k++); k; } \\ Michel Marcus, Sep 17 2014

More terms from Peter J. C. Moses, Sep 14 2014

A247342 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the odd part (A000265) of concatenation b_k 2^n is prime, or a(n)=0 if there is no such prime.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 4, 3, 2, 1, 3, 1, 1, 6, 1, 1, 1, 3, 1, 15, 29, 5, 1, 2, 3, 6, 1, 6, 20, 6, 3, 50, 3, 22, 8, 5, 5, 1, 84, 8, 7, 36, 3, 6, 7, 20, 6, 6, 8, 1, 6, 3, 2, 38, 1, 5, 3, 2, 5, 16, 1, 12, 13, 7, 1, 4, 16, 5, 32, 1, 6, 13, 4, 150, 7, 29, 17, 9, 12, 34

Offset: 0

Vladimir Shevelev, Sep 14 2014

Conjecture: for all n, a(n)>0.

a(443) > 17000 if it is not 0.

			2^0=1 and already 31 is prime. So a(0)=1;
2^1=2, but odd part of 32 is 1 (nonprime); then consider odd part of 332. It is 83 that is prime. So a(1)=2.

Robert Israel, Table of n, a(n) for n = 0..442

Cf. A000079, A232210, A242775, A247341.

f:= proc(n) local m,d,k,x;
    m:= 2^n;
    d:=ilog10(m);
    for k from 1 do
       x:= (10^k-1)/3*10^(d+1)+m;
       if isprime(x/2^padic:-ordp(x,2)) then return k fi
    od
end proc:
map(f, [$0..100]); # Robert Israel, Oct 30 2016

a(n) = {k = 0; while (! ((val = eval(concat(Str((10^k-1)/3), Str(2^n)))) && isprime(val/2^valuation(val, 2))), k++); k;} \\ Michel Marcus, Sep 15 2014

More terms from Michel Marcus, Sep 15 2014

A245657 Primes p for which none of the concatenations p3, p9, 3p, 9p are primes.

Original entry on oeis.org

3, 107, 113, 179, 317, 443, 487, 599, 641, 653, 751, 773, 937, 977, 991, 1021, 1087, 1103, 1187, 1201, 1213, 1217, 1301, 1409, 1427, 1439, 1483, 1553, 1559, 1579, 1609, 1637, 1693, 1747, 1777, 1787, 1789, 1861, 1949, 1987, 1993, 2081, 2129, 2239, 2281, 2287, 2293, 2351, 2393, 2477

Offset: 1

Vladimir Shevelev, Sep 13 2014

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A232210, A242775, A247341, A247342.

Select[Prime[Range[400]],NoneTrue[{10#+3,10#+9,3*10^IntegerLength[#]+#, 9*10^IntegerLength[ #]+#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 06 2020 *)

lista(nn) = {forprime(p=2, nn, if (!isprime(eval(concat(Str(p), Str(3)))) && ! isprime(eval(concat(Str(p), Str(9)))) && ! isprime(eval(concat(Str(3), Str(p)))) && ! isprime(eval(concat(Str(9), Str(p)))), print1(p, ", ")););} \\ Michel Marcus, Sep 14 2014

import sympy
from sympy import isprime
from sympy import prime
for n in range(1,10**3):
  p = str(prime(n))
  if not isprime(int(p+'3')) and not isprime(int(p+'9')) and not isprime(int('3'+p)) and not isprime(int('9'+p)):
    print(int(p),end=', ') # Derek Orr, Sep 16 2014

More terms from Derek Orr, Sep 16 2014

cp's OEIS Frontend

A232210 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A242775 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation b_k and prime(n) is prime, or a(n)=0 if there is no such prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A247342 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the odd part (A000265) of concatenation b_k 2^n is prime, or a(n)=0 if there is no such prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A245657 Primes p for which none of the concatenations p3, p9, 3p, 9p are primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions