A047777 -id:A047777 - cp's OEIS frontend

A195834 Increasing primes formed from the decimal expansion of Pi, contiguous and smallest.

3, 14159, 26535897932384626433832795028841971693993751058209, 74944592307816406286208998628034825342117067982148086513282306647093

Offset: 1

Omar E. Pol, Oct 21 2011

Leading zero not allowed thus forcing continuation of previous term. All digits of pi are in the concatenion of all terms of this sequence.

Cf. A000796, A005042, A047777, A050819.

a(n) < a(n+1).

More terms from D. S. McNeil, Oct 22 2011

A050817 Odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

Original entry on oeis.org

3, 1, 41, 5, 9, 265, 35, 89, 7, 93, 23, 8462643, 383, 27, 95028841, 97, 169, 39, 937, 5105, 8209, 749, 445, 92307, 81, 64062862089, 9862803, 4825, 3421, 17067, 9821, 480865, 13, 28230664709, 3844609, 5505, 8223, 17, 25, 359, 4081, 28481, 11

Offset: 1

Patrick De Geest, Oct 15 1999

Leading zero not allowed thus forcing continuation of previous term.

C. Rivera, Problem 18, The Prime Puzzles & Problems Connection.

Cf. A047777, A050818, A016062, A050819, A050807.

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", "))))

A229155 Number of digits of the n-th term of the decimal expansion of e = exp(1) cut into chunks of primes.

Original entry on oeis.org

1, 1, 649, 1, 1, 2, 29, 1, 1, 2, 1, 1, 1, 53, 1872, 3, 5

Offset: 1

M. F. Hasler, Sep 15 2013

Trying to cut the decimal expansion A001113 of e=2.718281828... into "prime chunks", one gets (2, 7, p, 5, 3, 11, q, 7, 3, 61, 3, 3, 2, r, ...) where p, q, r are 649-, 29-, 53-digit primes, respectively. The size of p makes it impossible to register this more fundamental sequence in the OEIS as it is done in A047777 for Pi. This led us to store just the length of the terms in this sequence.

Sequence A121267 is a (not exact) analog for Pi; note that A047777 requires all primes to be distinct, while we allow repetition of 7, 3, 2, ... as seen in the above example. If we did not, the terms following 29 would be 2, 2, 6, 3, 7, 8, 3, 441, 9, 17, ... instead of 1, 1, 2, 1, 1, 1, 53, ...

Joseph L. Pe, Trying to Write e as a Concatenation of Primes (2009)

Cf. A007512, A001113, A047658, A064118.

default(realprecision,2000);c=exp(1)/10;for(k=1,9e9,ispseudoprime(c\.1^k) & !print1(k,",") & k=0*c=frac(c*10^k))

a(15)-a(17) from Jinyuan Wang, Mar 26 2020

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

A229178 Primes seen in decimal expansion of e, contiguous, smallest and distinct, with terms larger than 10^100 replaced by 0.

Original entry on oeis.org

2, 7, 0, 5, 3, 11, 80232878250981945581530175671, 73, 61, 332069, 811, 2509961, 81881593, 41, 0, 30592123, 66771943252786753, 0, 0, 6343, 490769, 64237, 2229435236612557250881, 47, 79, 223, 151, 97, 477806056967, 253801, 71, 0, 0, 9467, 83, 48197

Offset: 1

M. F. Hasler, Sep 15 2013

Sequence A047777 is the analog for Pi.

Without the admittedly arbitrary clause "terms larger than ... replaced by zero", the sequence could not be stored beyond its second term, since the 3rd term would have 649 digits (and the 15th term would have 441 digits). Although this restriction is arbitrary, several other (maybe more natural) alternatives (for example, larger than the concatenation of the preceding/following 10 terms...) would yield the same initial terms.

Cf. A001113, A007512, A047777, A064118, A229155.

default(realprecision,5000);c=exp(1)/10;u=[];for(k=0,9e9,ispseudoprime(c\.1^k) & !setsearch(u,c\.1^k) & (u=setunion(u,Set(c\.1^k))) & !print1(c\.1^k,",") & k=0*c=frac(c*10^k))

More terms from Jinyuan Wang, Mar 26 2020

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A195834 Increasing primes formed from the decimal expansion of Pi, contiguous and smallest.

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A050817 Odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

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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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PARI

A229155 Number of digits of the n-th term of the decimal expansion of e = exp(1) cut into chunks of primes.

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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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A229178 Primes seen in decimal expansion of e, contiguous, smallest and distinct, with terms larger than 10^100 replaced by 0.

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PARI

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