A198019 -id:A198019 - cp's OEIS frontend

A005042 Primes formed by the initial digits of the decimal expansion of Pi.

3, 31, 314159, 31415926535897932384626433832795028841

Offset: 1

The next term consists of the first 16208 digits of Pi and is too large to show here (see A060421). Ed T. Prothro found this probable prime in 2001.

A naive probabilistic argument suggests that the sequence is infinite. - Michael Kleber, Jun 23 2004

M. Gardner, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Gardner, Letter to N. J. A. Sloane, Nov 16 1979.
Ed T. Prothro, How I Found the Next Pi Prime.
Eric Weisstein's World of Mathematics, Pi-Prime.
Index entries for sequences related to "constant primes"
Index entries for sequences related to the number Pi

See A060421 for further terms.

Cf. A198018, A198019, A195834, A047777, A053013, A064467.

Digits := 130; n0 := evalf(Pi); for i from 1 to 120 do t1 := trunc(10^i*n0); if isprime(t1) then print(t1); fi; od:

a = {}; Do[k = Floor[Pi 10^n]; If[PrimeQ[k], AppendTo[a, k]], {n, 0, 160}]; a (* Artur Jasinski, Mar 26 2008 *)
nn=1000;With[{pidigs=RealDigits[Pi,10,nn][[1]]},Select[Table[FromDigits[ Take[pidigs,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Sep 26 2012 *)

c=Pi;for(k=0,precision(c),isprime(c\.1^k) & print1(c\.1^k,",")) \\ - M. F. Hasler, Sep 01 2013

a(n) = floor(10^(A060421(n)-1)*A000796), where A000796 is the constant Pi = 3.14159... . - M. F. Hasler, Sep 02 2013

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

Original entry on oeis.org

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

Offset: 1

Carlos Rivera

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

A198018 Yet unseen primes occurring within the first 1,2,3,4,... digits of Pi, A000796 (ordered according to position of last, then initial digit).

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Oct 20 2011

Consider the first, then the first two, then the first three, ..., terms of A000796, i.e., decimal digits of Pi. Look whether the concatenation of a certain number of subsequent digits yields a prime which did not yet occur earlier (and thus necessarily involves the last of the considered digits). If so, add this prime to the sequence.

Contains A005042 as a subsequence.

			The first digit of the sequence is the prime a(1)=3.
The first two digits, "3.1", yield the prime a(2)=31.
In "3.14" there are no more primes. In "3.141" there is the prime a(3)=41.
In "3.1415" there is the prime a(4)=5.
In "3.14159" we have the primes 314159, 14159, 4159 and 59.

Cf. A198019 ("new" primes ordered w.r.t. their size instead of starting position).

Cf. A000796, A005042, A047658.

{my(PI=digits(Pi\.1^30), seen=[]); for(i=1, #PI-1, for(j=1, i, my(p=fromdigits(PI[j..i])); !isprime(p) || setsearch(seen, p) || print1(p", ") || seen=setunion(seen,[p])))} \\ edited to use current PARI syntax by Andrew Howroyd and M. F. Hasler, May 10 2021

{my(a=List()); for(m=0, precision(.)-3, my(pi=Pi\.1^m, p); for(k=0, m, !isprime(p=pi%10^(m-k+1)) && setsearch(Set(a), p) && listput(a,p))); a} \\ M. F. Hasler, May 10 2021

A198187 Primes from the decimal expansion of Pi, sorted first by the final digit index and then by length.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Oct 21 2011

In this sequence, primes are listed each time they occur (again) with a new ending position, in contrast to A198019 where only the first occurrence of each prime is listed. - M. F. Hasler, Sep 02 2013

			The first digit is 3, which is prime, so a(1) = 3.
The second digit is 1, which is no prime, but 31 is prime, so a(2) = 31.
The third digit is 4, which does not end any prime.
The fourth digit is 1, not prime, but 41 is prime, so a(3) = 41.

Cf. A104841.

Cf. A198018, A198019, A195834, A005042, A047777, A053013, A064467.

v=[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3]
for(n=1,#v,x=0;p=1;forstep(k=n,1,-1,x+=p*v[k];p*=10;if(v[k]&&isprime(x),print1(x", "))))

A226943 Semiprimes in the order in which they appear in the decimal expansion of Pi.

Original entry on oeis.org

4, 14, 314, 141, 15, 415, 1415, 9, 159, 6, 26, 926, 5926, 15926, 65, 265, 2653, 92653, 592653, 35, 535, 6535, 5926535, 58, 358, 265358, 314159265358, 589, 3589, 53589, 2653589, 92653589, 1592653589, 1415926535897, 979, 5358979, 59265358979, 159265358979

Offset: 1

Jonathan Vos Post, Sep 01 2013

This is to semiprimes A001358 as A198019 is to primes A000040. Considering the first 1, 2, 3, 4, ... digits of the decimal expansion 3.14159... of Pi, record the semiprimes that have not occurred earlier, the smaller first if two or more appear by the n-th digit that have not been seen in the first n-1 digits.

			There are no semiprimes in the first 1 or 2 digits (3, 31). Then after 3 digits we have three: 4, 14, and 314 appearing for the first time. So a(1) = 4, a(2) = 14 and a(3) = 314.

Giovanni Resta, Table of n, a(n) for n = 1..300

Cf. A000796, A001358, A198019.

semiQ[n_] := Total[Last /@ FactorInteger@n ] == 2; sp = Select[Range@ 999, semiQ]; spQ[n_] := If[n < 10^6, semiQ@n, ! Or @@ IntegerQ /@ (n/sp) && semiQ@ n]; seq = {}; Do[seq = Join[seq, Select[Union@ Complement[ Mod[FromDigits@ RealDigits[Pi, 10, n][[1]], 10^Range[n, 1, -1]], seq], spQ]], {n, 30}]; seq (* Giovanni Resta, Oct 01 2013 *)

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

A005042 Primes formed by the initial digits of the decimal expansion of Pi.

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

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A198018 Yet unseen primes occurring within the first 1,2,3,4,... digits of Pi, A000796 (ordered according to position of last, then initial digit).

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A198187 Primes from the decimal expansion of Pi, sorted first by the final digit index and then by length.

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A226943 Semiprimes in the order in which they appear in the decimal expansion of Pi.

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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.

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