A065987 -id:A065987 - cp's OEIS frontend

A004601 Expansion of Pi in base 2 (or, binary expansion of Pi).

1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 2

N. J. A. Sloane

			11.0010010000111111011010101000100010000...

J. P. Delahaye, Le Fascinant Nombre Pi, "100000 digits of pi in base two", pp. 209-210; Pour la Science, Paris 1997.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Francisco Javier Aragón Artacho, 2 billion step walk on the digits of pi
A. Brouty, Les décimales de pi en base 2 jusqu'à 1 million
Elias's Pi Page, Binary representation of pi with 32768 digits
J. Leroy, M. Rigo, and M. Stipulanti, Behavior of Digital Sequences Through Exotic Numeration Systems, Electronic Journal of Combinatorics 24(1) (2017), #P1.44. See w(n) in Example 19.
Steve Pagliarulo, Stu's pi page: base 2 (23 pages) [link was dead, retrieved from the Internet archive]

Cf. A119017, A068425, A117721, A065987, A051480, A007514.

Pi in base b: this sequence (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

convert(evalf(Pi), binary, 120);  # Alois P. Heinz, Dec 16 2018

RealDigits[Pi, 2, 75][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 2], {n, 1, 100}] (* Joan Ludevid, Jun 24 2022;easy to compute a(10000000)=0 with this function; requires Mathematica 12.0+ *)

PARI
```
binary(Pi) \\ Altug Alkan, Apr 08 2018
```

A060421 Numbers k such that the first k digits of the decimal expansion of Pi form a prime.

Original entry on oeis.org

1, 2, 6, 38, 16208, 47577, 78073, 613373

Offset: 1

Michel ten Voorde, Apr 05 2001

The Brown link states that in 2001 Ed T. Prothro reported discovering that 16208 gives a probable prime and that Prothro verified that all values for 500 through 16207 digits of Pi are composites. - Rick L. Shepherd, Sep 10 2002

The corresponding primes are in A005042. - Alexander R. Povolotsky, Dec 17 2007

			3 is prime, so a(1) = 1; 31 is prime, so a(2) = 2; 314159 is prime, so a(3) = 6; ...

K. S. Brown, Primes in the Decimal Expansion of Pi [Broken link?]
K. S. Brown, Primes in the Decimal Expansion of Pi [Cached copy]
Prime Curios, 314159
Prime Curios, 31415...36307 (16208-digits)
Shyam Sunder Gupta, Mystery of pi, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 19, 473-497.
Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Pi Digits
Eric Weisstein's World of Mathematics, Pi-Prime

Cf. A000796 (Pi), A005042, A007523, A047658.
Primes in other constants: A064118 (e), A065815 (gamma), A064119 (phi), A118328 (Catalan's constant), A115377 (sqrt(2)), A119344 (sqrt(3)), A228226 (log 2), A228240 (log 10), A119334 (zeta(3)), A122422 (Soldner's constant), A118420 (Glaisher-Kinkelin constant), A174974 (Golomb-Dickman constant), A118327 (Khinchin's constant).
In other bases: A065987 (binary), A065989 (ternary), A065991 (quaternary), A065990 (quinary), A065993 (senary).

Do[If[PrimeQ[FromDigits[RealDigits[N[Pi, n + 10], 10, n][[1]]]], Print[n]], {n, 1, 9016} ]

a(6) = 47577 from Eric W. Weisstein, Apr 01 2006
a(7) = 78073 from Eric W. Weisstein, Jul 13 2006
a(8) = 613373 from Adrian Bondrescu, May 29 2016

A117721 Primes formed by the initial digits of the binary expansion of Pi.

Original entry on oeis.org

3, 6588397, 1686629713, 26986075409, 16703571626015105435307505830654230989, 13420802360424337830311681948440006481608388178854297901454212848703426437343610760199000777828079

Offset: 1

Carl R. White, Apr 13 2006

Pi primes to base 2 (for base 10 see A005042). This sequence is the list of prime members of A068425.

Cf. A004601 (the binary expansion), A005042, A068425, A065987 (where primes are).

A119017 Primes from binary expansion of Pi, another version. Starting with the first bit of the binary expansion, A004601 = 1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,... we move rightward until we encounter another 1. Since 11 (= 3 in decimal) is prime, we move to the next 1 and repeat the process.

Original entry on oeis.org

3, 73, 4639, 67, 3, 3, 3, 3, 3, 5, 3, 5, 5, 5, 17, 17, 1069, 5, 3, 5, 17, 3, 9099300883537, 17, 3, 5, 19, 3, 17, 19, 3, 17, 3, 19, 3, 17, 5, 17, 5, 3, 3, 257, 3, 5, 3, 3, 131, 3, 3, 19, 3, 5, 17, 37, 5, 1153, 1033, 73, 19, 3, 3, 16657, 17, 17, 5, 19, 3, 19, 3, 3, 3, 3, 19, 3, 17, 3, 3

Offset: 1

Russell Walsmith, Jul 23 2006

Records: 3, 73, 4649, 9099300883537, 37848784972821936516494858855515680431107854546647118951099098009925403829863969526043052181881, ..., . - Robert G. Wilson v, Jul 24 2006

			11 = 3
1001001 = 73
1001000011111 = 4639
1000011 = 67
11 = 3
11 = 3
11 = 3

Cf. A004601, A068425, A117721, A065987.

ps = First@RealDigits[Pi, 2, 10^3]; lst = {}; Do[k = 1; While[fd = FromDigits[ Take[ps, k], 2]; EvenQ@fd || ! PrimeQ@fd, k++ ]; AppendTo[lst, fd]; j = 1; While[ ps[[j]] != 1, j++ ]; ps = Drop[ps, j], {n, 77}]; lst (* Robert G. Wilson v, Jul 24 2006 *)

More terms from Robert G. Wilson v, Jul 24 2006

A333649 Numbers k such that the second k binary digits of Pi represent a prime (leading zeros allowed).

Original entry on oeis.org

3, 7, 41, 93, 166, 316, 1449, 6605, 10015, 13097, 16284, 19075, 35137, 70558, 128436

Offset: 1

Robert Israel, Mar 31 2020

Numbers k such that floor(2^(2*k-2)*Pi) mod 2^k is prime.
A random number of k binary digits has probability ~ constant/k of being prime, so heuristically we should expect the sequence to be infinite, but growing exponentially.
a(16) > 2*10^5. - Michael S. Branicky, Dec 17 2024

			a(2) = 7 is a term because the first 14 binary digits in Pi are 11.001001000011; the second 7 binary digits are 1000011, or 67 in decimal, which is prime.

Cf. A004601, A065987.

L:= floor(Pi*2^19998):
select(n -> isprime(floor(L*2^(2*n-20000)) mod 2^n), [$1..10000]);

default(realprecision, 10^5);
is(k) = ispseudoprime(floor(4^(k-1)*Pi)%2^k); \\ Jinyuan Wang, Mar 31 2020

a(9) from Jinyuan Wang, Mar 31 2020
a(10)-a(13) from Chai Wah Wu, Apr 06 2020
a(14)-a(15) from Michael S. Branicky, Dec 16 2024

A119377 Numbers k such that the next k binary digits of Pi are odd primes with no leading zeros.

Original entry on oeis.org

2787, 6, 7, 23, 2, 3, 3, 8, 2, 2, 2, 5, 8, 2, 18, 9, 10, 413, 8, 3, 2, 4019, 14, 4, 2, 2, 11, 21, 4, 2, 3, 6, 2, 11, 3, 5, 19, 2, 6, 2, 4, 32, 2, 56, 31, 6, 7, 7, 2, 32, 20, 9, 10, 900, 2, 2, 2, 97, 5, 2, 8, 64, 3, 13, 3, 2, 6, 7, 15, 3, 2666, 7, 8, 3, 14, 3, 2, 2, 6, 5, 92, 17, 31, 4, 241, 78, 3

Offset: 1

Robert G. Wilson v, Jul 24 2006

Partition the string of binary digits of Pi in such a way that each partition begins and ends with 1 (thus no leading or trailing zeros) and each such partition is prime.
Pi_2 = 1100100100001111110110101010001000100001011010001100001000..._2 (A004601).
If 2 is allowed as a member, then the sequence begins: 2787,2,5,6,2,2,2,39,5,8,2,18,9,10,2,153,2,6,2,18,7,7,12,2,2,2,2,....

			a(1) represents the binary number 1100100100...(2767 terms)...0100000011 which equals the decimal number 7339860347...(819 terms)...8308318467 which is a prime.
a(2) represents the binary number 101001 which equals the decimal number 41, a prime.

Cf. A004601, A068425, A117721, A065987, A119017.

ps = First@ RealDigits[Pi, 2, 12010]; lst = {}; Do[k = 1; While[fd = FromDigits[ Take[ps, k], 2]; EvenQ@fd || ps[[k + 1]] == 0 || !PrimeQ@fd, k++ ]; AppendTo[lst, k]; ps = Drop[ps, k], {n, 87}]; lst

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A004601 Expansion of Pi in base 2 (or, binary expansion of Pi).

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A060421 Numbers k such that the first k digits of the decimal expansion of Pi form a prime.

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A117721 Primes formed by the initial digits of the binary expansion of Pi.

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A333649 Numbers k such that the second k binary digits of Pi represent a prime (leading zeros allowed).

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Maple

PARI

Extensions

A119377 Numbers k such that the next k binary digits of Pi are odd primes with no leading zeros.

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Mathematica