

A047777


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


17




OFFSET

1,1


COMMENTS

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 3057digit 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 3057digit 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 14650digit 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 748digit prime, see the afile from J.F. Alcover. Sequence A229181 lists the size of these terms.  M. F. Hasler, Sep 15 2013, updated Jan 18 2019


LINKS

Table of n, a(n) for n=1..8.
JeanFrancois 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 bfile, uploaded as afile 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".


EXAMPLE

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


MATHEMATICA

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 (* JeanFrançois Alcover, Oct 16 2013 *)


PROG

(PARI) {default(realprecision, N=3500); x=Pi; S=a=[]; while(N > L=logint(p=floor(x), 10), L%200!Lprint1("/*"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


CROSSREFS

Cf. A053013, A000796.
Cf. A005042, A104841, A198018, A198019, A198187.
Sequence in context: A262651 A309246 A055379 * A195834 A124393 A116182
Adjacent sequences: A047774 A047775 A047776 * A047778 A047779 A047780


KEYWORD

nice,nonn,base


AUTHOR

Carlos Rivera


EXTENSIONS

The next term is the 3057digit 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


STATUS

approved



