A195834 -id:A195834 - 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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

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