A198018 -id:A198018 - 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

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",")))}

A072952 Primes obtained as initial segments of the decimal expansion of the Euler-Mascheroni constant gamma = 0.5772... .

Original entry on oeis.org

5, 577, 5772156649015328606065120900824024310421

Offset: 1

Shyam Sunder Gupta, Aug 12 2002

The next term (a(4)) has 185 digits and is too large to include. - Harvey P. Dale, May 14 2013
Sequence A065815 gives the number of digits of a(n), resp. numbers k such that a(n) = floor(gamma*10^k). Sequences A005042, A007512, A115453, A119343, A210704, ... are the analog of the present sequence for Pi, e, sqrt(2), sqrt(3), 3^(1/3), ... - M. F. Hasler, Aug 31 2013
The original wording of the definition (and example) was "primes found in the decimal expansion..." which could as well refer to the sequence (5,7,7,215664901,5,3,2, ...) or (5,7,72156649, ...) or (5,7,7215664901, ...) (analogs to A047777 or A195834), or to the sequence (5,7,57, ...), analog to A198018. - M. F. Hasler, Sep 01 2013

			a(2) = 577, since 577 is the second prime obtained as initial segment of the decimal expansion of Euler-Mascheroni constant gamma = 0.577215664... .

Amiram Eldar, Table of n, a(n) for n = 1..4
Eric Weisstein's World of Mathematics, Constant Primes.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Digits.
Index entries related to "constant primes".

Cf. A001620, A047777, A065815, A195834, A198018.

Analogous sequences: A005042 (Pi), A007512 (e), A115453 (sqrt(2)), A119343 (sqrt(3)), A210704 (3^(1/3)).

nn=200;With[{emc=RealDigits[EulerGamma,10,nn][[1]]},Select[Table[ FromDigits[ Take[emc,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, May 14 2013 *)

default(realprecision, 777); /* use that many digits */
A072952={(c=Euler, v=1/*set to 0 for indices (i.e., A065815) instead of values*/)->for(k=0, precision(c), ispseudoprime(p=c\.1^k)&&print1([k, p][1+v]", "))} \\ M. F. Hasler, Aug 31 2013

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", "))))

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

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

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A072952 Primes obtained as initial segments of the decimal expansion of the Euler-Mascheroni constant gamma = 0.5772... .

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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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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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