A198187 -id:A198187 - cp's OEIS frontend

A047777 Primes seen in the decimal expansion of Pi (disregarding the decimal point) that are contiguous, smallest and distinct.

3, 14159, 2, 653, 5, 89, 7, 9323

Offset: 1

Sequence A121267 gives the number of digits of a(n) [but see also A229181 for a variant, cf. below]. The terms a(9)-a(11) had been found by Chris Nash in October 1999, and primality of the 3057-digit term a(9) has been proved in September 2002 by J. K. Andersen, who also found the next 5 terms a(12)-a(16) and the bound a(17) > 10^32000, cf. Rivera's web page "Problem 18". - M. F. Hasler, Aug 31 2013

There is a natural variant of the present sequence, using the same definition except for not requiring that all primes have to be distinct. That variant would have the same 3057-digit prime as next term a(9), and therefore have the same displayed terms and not justify a separate entry in the OEIS. However, terms beyond a(9) would be different: instead of a(10) = 73, a(11) = 467 and the 14650-digit PRP a(11), it would be followed by a'(10) = 7, a'(11) = 3 (which cuts a(10) = 73 in two pieces), a'(12) = 467, a'(13) = a'(14) = 2, and a'(15) equal to a 748-digit prime, see the a-file from J.-F. Alcover. Sequence A229181 lists the size of these terms. - M. F. Hasler, Sep 15 2013, updated Jan 18 2019

			The first digit of Pi = 3.14159... is the prime 3, therefore a(1) = 3.
We discard this digit 3, and look for the first time a chunk of subsequent digits (always starting with the 1 coming right after the previously used 3) would be prime: 1, 14, 141, 1415 are not, but 14159 is. (The single-digit prime '5' was not considered, because we require the primes made from the whole contiguous chunk of digits starting after the previously found prime.) Thus, a(2) = 14159.
Thereafter, we have the single-digit prime a(3) = 2, and then a(4) = 653 (since neither 6 nor 65 is prime). - _M. F. Hasler_, Jan 18 2019

Jean-Francois Alcover, Table of n, v(n) for n = 1..100 for the variant with duplicates described in the comment. Initially submitted on Oct 16 2013 as b-file, uploaded as a-file by _Georg Fischer_, Jan 18 2019
Joseph L. Pe, Trying to Write e as a Concatenation of Primes (2009) [from Internet Archive Wayback Machine]
Carlos Rivera, Problem 18. Pi as a concatenation of the smallest contiguous different primes, The Prime Puzzles and Problems Connection.
Index entries related to "constant primes".

Cf. A053013, A000796.

Cf. A005042, A104841, A198018, A198019, A198187.

digits = Join[{{3}}, RealDigits[Pi, 10, 4000] // First // Rest]; used = {}; primes = digits //. {a:({Integer..}..), b__Integer /; PrimeQ[p = FromDigits[{b}]] && FreeQ[used, p], c___Integer} :> (Print[p]; AppendTo[used, p]; {a, {p}, c}); Select[primes, Head[#] == List &] // Flatten (* _Jean-François Alcover, Oct 16 2013 *)

{default(realprecision,N=3500); x=Pi; S=a=[]; while(N > L=logint(p=floor(x),10), L%200||!L||print1("/*"L"*/"); if( ispseudoprime(p) && !setsearch(S,p), S=Set(a=concat(a,p)); print1(p","); x-=p; N-=logint(p,10)); x*=10); default(realprecision,38); a} \\ Remove the condition "&& !setsearch(S,p)" to get the variant allowing repetitions. The instruction "L%200..." is a progress indicator; it can be safely removed. - M. F. Hasler, Jan 18 2019

The next term is the 3057-digit prime formed from digits 19 through 3075. It is 846264338327950...708303906979207. - Mark R. Diamond, Feb 22 2000

The two terms after that are 73 and 467. - Jason Earls, Apr 05 2001

A198019 Primes occurring in the decimal expansion of Pi (A000796), ordered by position of last digit, then by size.

Original entry on oeis.org

3, 31, 41, 5, 59, 4159, 14159, 314159, 2, 53, 653, 1592653, 89, 141592653589, 7, 97, 5897, 35897, 6535897, 5926535897, 415926535897, 79, 58979, 358979, 589793, 23, 9323, 9265358979323, 43, 643, 462643, 93238462643, 433, 83, 383, 2643383, 38462643383, 89793238462643383, 41592653589793238462643383

Offset: 1

M. F. Hasler, Oct 20 2011

Cf. A198018; the only difference is that here we list the "new primes" by increasing size (for a given subsequence of A000796).
Considering the first 1, 2, 3, 4,.... digits of the decimal expansion 3.14159... of Pi, record the primes that have not occurred earlier.
Sequence A198187 lists "duplicate" primes multiple times, each time they occur anew ending in another decimal place. - M. F. Hasler, Sep 01 2013

			In Pi = 3... we have the prime a(1)=3.
In Pi = 3.1.... we have the prime a(2)=31.
In Pi = 3.14... we have no new prime.
In Pi = 3.141.... we have the prime a(3)=41.
In Pi = 3.1415.... we have the new prime a(5)=5.
In Pi = 3.14159.... we have the new primes (listed in increasing order) a(6)=59, a(7)=4159, a(8)=14159 and a(9)=314159. [_M. F. Hasler_, Sep 01 2013]

Cf. A000796, A198018.

Cf. A047777, A053013, A064467.

{t=Pi; u=[]; for(i=0,precision(t), for(k=1,i+1, ispseudoprime(p=t\.1^i%10^k)& !setsearch(u,p)& (u=setunion(u,Set(p)))&print1(p",")))}

A245571 a(n) is the smallest prime number with at least two digits formed by the concatenation of the subsequent digits of Pi, starting at the n-th digit, ignoring the decimal point.

Original entry on oeis.org

31, 14159, 41, 1592653, 59, 9265358979323, 26535897932384626433832795028841971693993751058209, 653, 53, 35897, 5897, 89, 97, 79, 9323, 32384626433832795028841971693993751058209749445923078164062862089986280348253421, 23, 38462643383

Offset: 1

Felix Fröhlich, Aug 22 2014

a(19) has 3057 digits. - Robert Israel, Aug 27 2014
a(20) = 462643. - Felix Fröhlich, Aug 30 2014
a(21) has >= 3490 digits, a(22) = 2643383, a(22)-a(42) have 20 or fewer digits. - Chai Wah Wu, Sep 24 2014

			a(4) = 1592653, because starting at the 4th digit in the expansion, the smallest substring of the digits of Pi forming a prime number is 3.14|1592653|589...

Cf. A000796, A005042, A047777, A076094, A104841, A195834, A198018, A198019, A198187.

N:= 1000: # to use up to N+1 digits of pi.
nmax:= 30: # to get up to a(nmax), if possible.
S:= floor(10^N*Pi):
L:= ListTools:-Reverse(convert(S,base,10)):
for n from 1 to nmax do
  p:= L[n];
  for k1 from n+1 to N+1 do
    p:= 10*p + L[k1];
    if isprime(p) then break fi
  od:
  if k1 > N+1 then
    A[n]:= "Ran out of digits";
    break
   else
    A[n]:= p
  end
od:
seq(A[i],i=1..n-1); # Robert Israel, Aug 27 2014

from sympy.mpmath import *
from sympy import isprime
def A245571(n):
    mp.dps = 1000+n
    s = nstr(pi,mp.dps)[:-1].replace('.','')[n-1:]
    for i in range(len(s)-1):
        p = int(s[:i+2])
        if p > 10 and isprime(p):
            return p
    else:
        return 'Ran out of digits'
# Chai Wah Wu, Sep 16 2014, corrected Chai Wah Wu, Sep 24 2014

cp's OEIS Frontend

A047777 Primes seen in the decimal expansion of Pi (disregarding the decimal point) that are contiguous, smallest and distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A198019 Primes occurring in the decimal expansion of Pi (A000796), ordered by position of last digit, then by size.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A245571 a(n) is the smallest prime number with at least two digits formed by the concatenation of the subsequent digits of Pi, starting at the n-th digit, ignoring the decimal point.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Python