A051480 -id:A051480 - 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
```

A256108 Positions of nonzero digits in binary expansion of Pi.

Original entry on oeis.org

-1, 0, 3, 6, 11, 12, 13, 14, 15, 16, 18, 19, 21, 23, 25, 29, 33, 38, 40, 41, 43, 47, 48, 53, 57, 58, 60, 63, 64, 68, 71, 72, 76, 77, 80, 81, 85, 87, 91, 93, 94, 95, 103, 104, 106, 107, 108, 114, 115, 116, 119, 120, 122, 126, 129, 131, 134, 141, 144, 147, 148, 149, 155, 159

Offset: 1

David S. Metzler, Mar 14 2015

Nonzero entries in A004601 (re-indexed to start at -1 and ascend).

The binary positions (exponents) are negated for convenience (as is standard practice). By the results of the PiHex project, the number 1,000,000,000,000,060 (for example) eventually appears in this sequence. Submitted on 3/14/15, (decimal) Pi Day.

			The most significant nonzero binary digit of pi occurs in the 2^1 position. Then there is a digit in the 2^0 position, then the 2^(-3) position, etc. Negate the exponents appearing to get this sequence.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Steve Pagliarulo, Stu's pi page: base 2
Colin Percival, PiHex Home Page
Wikipedia, PiHex

Cf. A004601 (Pi in base 2), A051480.

PositionIndex[First[RealDigits[Pi, 2, 200]]][1] - 2 (* Paolo Xausa, Aug 04 2024 *)

A256108_upto(N)={localbitprec(N+20); [i-2|i<-[1..-20+#N=concat(binary(Pi))], N[i]]} \\ M. F. Hasler, Jul 27 2024

Pi = Sum_{n>=0} 2^(-a(n)).

This sequence A256108 = { i | A004601(1-i) = 1 }. - M. F. Hasler, Jul 27 2024

A121657 Successive run lengths in decimal expansion of Pi.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Rick L. Shepherd, Aug 13 2006

			a(25) = 2 as the 25th run, the first run of two or more identical digits in 3.14159265358979323846264338..., consists of the two 3s following the first 24 digits.

Cf. A000796, A051480, A082586 (another version).

Length /@ Split@RealDigits[Pi, 10, 111][[1]] (* Robert G. Wilson v, Aug 16 2006 *)

A373471 Primes indices of nonzero bits in Pi (A256108).

Original entry on oeis.org

3, 11, 13, 19, 23, 29, 41, 43, 47, 53, 71, 103, 107, 131, 149, 163, 173, 179, 197, 211, 227, 229, 239, 269, 281, 283, 293, 311, 349, 373, 379, 397, 409, 421, 457, 541, 557, 563, 577, 587, 599, 601, 607, 613, 617, 643, 647, 653, 659, 673, 709, 727, 733, 739, 743

Offset: 1

M. F. Hasler, Jul 27 2024

Base 2 analog of A088565 (prime positions of '1's in decimal digits of Pi).

			Written in binary, Pi = 11.0010010000111111011010101000100010...[2] = Sum_{n >= -1} 2^-A256108(n), so the bits 1 have positions (-1, 0, 3, 6, 11, 12, 13, 14, 15, 16, 18, 19, 21, 23, 25, 29, 33, ...) and primes in this sequence are (3, 11, 13, 19, 23, ...) = this sequence.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A256108 (indices of nonzero bits in Pi), A004601 (Pi in base 2), A051480 (run lengths in A004601).
Cf. A088563 (indices of '0's in decimals of Pi).
Cf. A088565 (prime indices of '1's in the decimal digits of Pi).

Select[PositionIndex[First[RealDigits[Pi, 2, 1000]]][1] - 2, PrimeQ] (* Paolo Xausa, Jul 31 2024 *)

PARI
```
select(isprime, A256108_upto(777))
```

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

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A256108 Positions of nonzero digits in binary expansion of Pi.

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A121657 Successive run lengths in decimal expansion of Pi.

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Mathematica

A373471 Primes indices of nonzero bits in Pi (A256108).

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Mathematica

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