A232013 - OEIS

%I #34 Oct 19 2024 15:57:32

%S 4,1,12,4,13,14,11,10

%N Number of iterations of A176341 ("position of n in Pi") until a value is reached for the second time, when starting with n, or -1 if no value is repeated.

%C See A232014 for a variant based on A032445 instead of A176341.

%C Some loops: (1), (711939213), (0, 32, 15, 3), (19, 37, 46), (40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169).

%C See Hans Havermann table (in links) for primary unknown-length evolutions.

%H David G. Andersen, <a href="http://www.angio.net/pi">Loop Sequences within Pi, on The Pi-Search Page</a> (Search 2*10^8 decimal digits of Pi).

%H Hans Havermann, <a href="http://chesswanks.com/seq/a232013.txt">Information table of n, a(n) for n=0..100</a>.

%H Joaquin Navarro, <a href="http://images-archive.math.cnrs.fr/Les-secrets-du-nombre-Pi.html">Les secrets du nombre Pi</a> (Book review, in French).

%H James Taylor, <a href="http://www.subidiom.com/pi/">Irrational Numbers Search Engine</a> (Search 2*10^9 decimal digits of Pi).

%H Ady Tzidon, <a href="http://perplexus.info/show.php?pid=7746&amp;op=sol">Loops in Pi</a>.

%e a(0)=4 since A176341(0)=32 (position of the first "0" in Pi's digits), A176341(32)=15 (position of the first "32" in Pi's digits), A176341(15)=3 (position of the first "15" in Pi's digits), A176341(3)=0 (position of the first "3" in Pi's digits); here we find the "0" again after 4 iterations, thus a(0)=4.

%e a(1)=1 since A176341(1)=1 (the first "1" occurs at position 1 in Pi's digits), which already "closes the loop" after 1 iteration.

%e a(2)=12 because the iterations yield 2 > 6 > 7 > 13 > 110 > 174 > 155 > 314 > 0 > 32 > 15 > 3 > 0, here we re-enter the loop (of length 4) after 12 iterations.

%t pidigits = First[RealDigits[N[Pi, 10^6]]];

%t Table[ lst = {}; test = n; steps = 1;

%t While[AppendTo[lst, test]; !

%t    MemberQ[lst,

%t     test = First[

%t        First[SequencePosition[pidigits, IntegerDigits[test], 1]]] - 1],

%t steps++ ]; steps, {n, 0, 7}] (* _Robert Price_, Aug 31 2019 *)

%o (PARI) A232013(n)={my(u=0);for(i=1,9e9,u+=1<<n;bittest(u,n=A176341(n))&&return(i))}

%K nonn,base,more

%O 0,1

%A _M. F. Hasler_, Nov 16 2013

%E Edited by _Hans Havermann_, Aug 01 2014