A232013 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.

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

Offset: 0

Author

M. F. Hasler, Nov 16 2013

Keywords

• nonn
• base
• more

Comments

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

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).

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

Examples

			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.
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.
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.
		

Links

• David G. Andersen, Loop Sequences within Pi, on The Pi-Search Page (Search 2*10^8 decimal digits of Pi).
• Hans Havermann, Information table of n, a(n) for n=0..100.
• Joaquin Navarro, Les secrets du nombre Pi (Book review, in French).
• James Taylor, Irrational Numbers Search Engine (Search 2*10^9 decimal digits of Pi).
• Ady Tzidon, Loops in Pi.

Programs

Mathematica

pidigits = First[RealDigits[N[Pi, 10^6]]];
Table[ lst = {}; test = n; steps = 1;
While[AppendTo[lst, test]; !
   MemberQ[lst,
    test = First[
       First[SequencePosition[pidigits, IntegerDigits[test], 1]]] - 1],
steps++ ]; steps, {n, 0, 7}] (* Robert Price, Aug 31 2019 *)

PARI

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

Extensions

Edited by Hans Havermann, Aug 01 2014