A047777 -id:A047777 - cp's OEIS frontend

A121267 Number of decimal digits in A047777(n).

1, 5, 1, 3, 1, 2, 1, 4, 3057, 2, 3, 14650, 4, 2, 46, 4

Offset: 1

Robert G. Wilson v, Aug 23 2006

Not equal to the first differences of A053013 (which is based on a different variant of A047777).
"A new visitor to an old problem! Jens Kruse Andersen confirmed the primality of Nash's 3057-digit prime using PRIMO." - Carlos Rivera, Sep 2002
If it exists, then a(17) > 50000. - Lucas A. Brown, Jul 04 2024

Lucas A. Brown, Python program.
Carlos Rivera, Problem 18. Pi as a concatenation of the smallest contiguous different primes, The Prime Puzzles & Problems Connection.

Cf. A047777, A053013.

default(realprecision,5000);c=Pi/10;u=[];for(k=1,9e9,ispseudoprime(c\.1^k) & !setsearch(u,c\.1^k) & (u=setunion(u,Set(c\.1^k))) & !print1(k,",") & k=0*c=frac(c*10^k)) \\ M. F. Hasler, Sep 15 2013

The old values a(10) onwards were incorrect, as discovered by M. F. Hasler, Aug 31 2013. The old values, the b-file and the Mathematica program have been deleted. - N. J. A. Sloane, Sep 01 2013

A229181 Number of decimal digits in the variant of A047777(n) (decimal expansion of Pi cut in "prime chunks") without the restriction that all primes must be different.

Original entry on oeis.org

1, 5, 1, 3, 1, 2, 1, 4, 3057, 1, 1, 3, 1, 1, 748, 2, 2, 1, 2, 83, 5, 1, 2, 71, 10, 1, 1, 2, 2, 2, 1, 1, 3, 1, 14, 2, 5, 51, 1, 6, 1, 6, 3, 2, 9, 1, 16, 2, 3, 43, 1, 6, 19, 1, 5, 3, 1999, 1, 1, 2, 22, 1, 3, 1, 2, 2, 1, 2, 2, 5, 1, 1, 1, 1, 4, 1, 1, 3, 7, 5, 1, 6, 4, 3, 1, 10, 7, 1, 2, 11, 2, 5, 1, 13, 1, 20, 16, 1, 9, 16

Offset: 1

M. F. Hasler, Sep 15 2013

A variant of A121267. First differences of A053013. See these two sequences for further details.

C. Rivera, Prime strings

Cf. A047777, A053013, A121267, A229155.

A229181 = {1}; digits = Join[{{1}}, RealDigits[Pi, 10, 10^4] // First // Rest]; digits //. {a:({A229181,%20lg%5D;%20%7B%7B1%7D,%20c%7D)%20;%20A229181%20(*%20_Jean-Fran%C3%A7ois%20Alcover">Integer..}..), b__Integer /; PrimeQ[FromDigits[{b}]], c___Integer} :> (Print[lg = {b} // Length]; AppendTo[A229181, lg]; {{1}, c}) ; A229181 (* _Jean-François Alcover, Oct 17 2013 *)

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

More terms from Jean-François Alcover, Oct 17 2013

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

Original entry on oeis.org

3, 31, 314159, 31415926535897932384626433832795028841

Offset: 1

N. J. A. Sloane

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

A053013 Positions in decimal expansion of Pi where next prime begins.

Original entry on oeis.org

1, 2, 7, 8, 11, 12, 14, 15, 19, 3076, 3077, 3078, 3081, 3082, 3083, 3831, 3833, 3835, 3836, 3838, 3921, 3926, 3927, 3929, 4000, 4010, 4011, 4012, 4014, 4016, 4018, 4019, 4020, 4023, 4024, 4038, 4040, 4045, 4096, 4097, 4103, 4104, 4110, 4113, 4115, 4124

Offset: 1

Mark R. Diamond, Feb 22 2000

This sequence refers to a variant of A047777 in which one does not require that all primes are distinct. See A229181 for first differences of this sequence. - M. F. Hasler, Sep 15 2013

A229181 = {1}; digits = Join[{{1}}, RealDigits[Pi, 10, 5000] // First // Rest]; digits //. {a:({A229181,%20lg%5D;%20%7B%7B1%7D,%20c%7D)%20;%20Join%5B%7B1%7D,%20Accumulate%5BA229181%5D%20+%201%5D%20(*%20_Jean-Fran%C3%A7ois%20Alcover">Integer..}..), b__Integer /; PrimeQ[FromDigits[{b}]], c___Integer} :> (Print[lg = {b} // Length]; AppendTo[A229181, lg]; {{1}, c}) ; Join[{1}, Accumulate[A229181] + 1] (* _Jean-François Alcover, Oct 17 2013 *)

Partial sums of A229181: a(n)=sum_{k=1...n} A229181(k). - M. F. Hasler, Sep 15 2013

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",")))}

A050819 Increasing odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

Original entry on oeis.org

3, 141, 59265, 358979, 3238462643, 3832795028841, 97169399375105, 820974944592307, 81640628620899862803, 48253421170679821480865, 132823066470938446095505, 82231725359408128481117450284102701, 9385211055596446229489549303819644288109

Offset: 1

Patrick De Geest, Oct 15 1999

Leading zero not allowed thus forcing continuation of previous term.

Carlos Rivera, Problem 18. Pi as a concatenation of the smallest contiguous different primes, The Prime Puzzles & Problems Connection.

Cf. A000796, A016062, A037244, A047777, A050817, A016062, A050818, A050807.

a(12) corrected and a(13) from Sean A. Irvine, Aug 19 2021

A074721 Concatenate the primes as 2357111317192329313..., then insert commas from left to right so that between each pair of successive commas is a prime, always making the new prime as small as possible.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 2, 93137414347535961677173798389971011031071091131, 2, 7, 13, 11, 3, 7, 13, 91491511, 5, 7, 163, 167, 17, 3, 17, 9181, 19, 11, 9319, 7, 19, 9211223227229233239241251257, 2, 6326927, 127, 7, 2, 81283, 2, 93307, 3, 11, 3, 13, 3, 17, 3, 3, 13, 3, 7, 3, 47, 3, 493533593673733

Offset: 1

Reinhard Zumkeller, Sep 04 2002

Note that leading zeros are dropped. Example: When the primes 691, 701, 709, and 719 get concatenated and digitized, they become {..., 6, 9, 1, 7, 0, 1, 7, 0, 9, 7, 1, 9, ...}. These will end up in A074721 as: a(98)=691, a(99)=7, a(100)=17, a(101)=97, a(102)=19, ..., . Terms a(100) & a(101) have associated with them unstated leading zeros. - Hans Havermann, Jun 26 2009
Large terms in the links are probable primes only. For example, a(1290) has 24744 digits and a(4050), 32676 digits. If of course any probable primes were not actual primes, the indexing of subsequent terms would be altered. - Hans Havermann, Dec 28 2010
What is the next term after {2, 3, 5, 7, 11, 13, 17, 19}, if any, giving a(k)=A000040(k)?

Robert G. Wilson v, Table of n, a(n) for n = 1..329 [a(330) is too large to be included in a b-file: see the a-file]
Hans Havermann, Two-color listing of 5359 terms
Robert G. Wilson v, Table of n, a(n) for n = 1..1289

Cf. A073034, A047777, A053648, A069090.

Cf. A033308.

a074721 n = a074721_list !! (n-1)
a074721_list = f 0 $ map toInteger a033308_list where
   f c ds'@(d:ds) | a010051'' c == 1 = c : f 0 ds'
                  | otherwise = f (10 * c + d) ds
-- Reinhard Zumkeller, Mar 11 2014

id = IntegerDigits@ Array[ Prime, 3000] // Flatten; lst = {}; Do[ k = 1; While[ p = FromDigits@ Take[ id, k]; !PrimeQ@p || p == 1, k++ ]; AppendTo[lst, p]; id = Drop[id, k], {n, 1289}]

a=0;
tryd(d) = { a=a*10+d; if(isprime(a),print(a);a=0); }
try(p) = { if(p>=10,try(p\10)); tryd(p%10); }
forprime(p=2,1000,try(p)); \\ Jack Brennen, Jun 25 2009

Edited by Robert G. Wilson v, Jun 26 2009

Further edited by N. J. A. Sloane, Jun 27 2009, incorporating comments from Leroy Quet, Hans Havermann, Jack Brennen and Franklin T. Adams-Watters

A050807 Increasing even numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

Original entry on oeis.org

314, 1592, 65358, 97932, 384626, 433832, 795028, 8419716, 939937510, 5820974944, 59230781640, 62862089986, 280348253421170, 679821480865132, 8230664709384460, 955058223172535940, 8128481117450284102, 70193852110555964462, 2948954930381964428810

Offset: 1

Patrick De Geest, Oct 15 1999

Leading zero not allowed thus forcing continuation of previous term.

Carlos Rivera, Problem 18. Pi as a concatenation of the smallest contiguous different primes, The Prime Puzzles & Problems Connection.

Cf. A000796, A047777, A050817, A016062, A050818, A050819.

More terms from Sean A. Irvine, Aug 19 2021

A050818 Even numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

Original entry on oeis.org

314, 1592, 6, 5358, 97932, 38, 4, 62, 64, 338, 32, 7950, 2, 8, 84, 19716, 939937510, 58, 20, 974, 94, 4592, 30, 78, 16, 40, 628, 620, 8998, 6280, 34, 82, 534, 21170, 6798, 214, 80, 86, 5132, 8230, 66, 470, 938, 44, 60, 9550, 582, 23172, 53594, 0, 812, 848

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, A050817, A016062, A050819, A050807.

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

cp's OEIS Frontend

A121267 Number of decimal digits in A047777(n).

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A229181 Number of decimal digits in the variant of A047777(n) (decimal expansion of Pi cut in "prime chunks") without the restriction that all primes must be different.

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Mathematica

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

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Maple

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A053013 Positions in decimal expansion of Pi where next prime begins.

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Mathematica

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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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PARI

A050819 Increasing odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

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A074721 Concatenate the primes as 2357111317192329313..., then insert commas from left to right so that between each pair of successive commas is a prime, always making the new prime as small as possible.

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Haskell

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A050807 Increasing even numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

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