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A047777 Primes seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct. 17
3, 14159, 2, 653, 5, 89, 7, 9323 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=1..8.

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".


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


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-Fran├žois Alcover, Oct 16 2013 *)


(PARI) {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


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




Carlos Rivera


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



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Last modified September 27 04:03 EDT 2022. Contains 357052 sequences. (Running on oeis4.)