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Positions of nonzero digits in binary expansion of Pi.
5

%I #17 Aug 04 2024 20:49:45

%S -1,0,3,6,11,12,13,14,15,16,18,19,21,23,25,29,33,38,40,41,43,47,48,53,

%T 57,58,60,63,64,68,71,72,76,77,80,81,85,87,91,93,94,95,103,104,106,

%U 107,108,114,115,116,119,120,122,126,129,131,134,141,144,147,148,149,155,159

%N Positions of nonzero digits in binary expansion of Pi.

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

%C 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.

%H Paolo Xausa, <a href="/A256108/b256108.txt">Table of n, a(n) for n = 1..10000</a>

%H Steve Pagliarulo, <a href="https://web.archive.org/web/20160324234131/http://members.shaw.ca/francislyster/pi/pistats/pibase2.pdf">Stu's pi page: base 2</a>

%H Colin Percival, <a href="https://web.archive.org/web/20150827020352/http://oldweb.cecm.sfu.ca/projects/pihex/">PiHex Home Page</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/PiHex">PiHex</a>

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

%F This sequence A256108 = { i | A004601(1-i) = 1 }. - _M. F. Hasler_, Jul 27 2024

%e 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.

%t PositionIndex[First[RealDigits[Pi, 2, 200]]][1] - 2 (* _Paolo Xausa_, Aug 04 2024 *)

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

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

%K sign,base

%O 1,3

%A _David S. Metzler_, Mar 14 2015