A286011 a(1)=1, and for n>1, a(n) is the maximum number of iterations of sigma resulting in n, starting at some integer k; or 0 if n cannot be reached from any k.

1, 0, 1, 2, 0, 1, 3, 4, 0, 0, 0, 2, 1, 2, 5, 0, 0, 1, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 1, 1, 2, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 1, 0, 0, 7, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2

Offset: 1

Author

Michel Marcus, Apr 30 2017

Keywords

• nonn

Comments

a(n)=0 for n in A007369 and a(n)>0 for n in A002191.

Records are found at indices given by A007497.

The above would be correct for a(1) = 0 (in a weak sense) or rather a(1) = -1 (for infinity), but as the sequence is defined, 2 & 3 do not produce a record, so the indices of records are 1, (3), 4, 7, ... = {1} U A007497 \ {2, (3)}. - M. F. Hasler, Nov 20 2019

Examples

			a(4)=2 because 4=sigma(3), but also sigma(sigma(2)) with 2 iterations.
a(7)=3 because 7=sigma(4), but also sigma(sigma(3)), and sigma(sigma(sigma(2))), with 3 iterations.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005

Crossrefs

Cf. A000203, A002191, A007369, A007497, A257670.

Programs

Maple

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
for n from 1 to N do
  s:= numtheory:-sigma(n);
  if s <= N then V[s]:= max(V[s],V[n]+1) fi
od:
convert(V,list); # Robert Israel, May 01 2017

PARI

a(n) = {if (n==1, return(1)); vn = vector(n-1, k, k+1); nb = 0; knb = 0; ok = 1; while(ok, nb++; vn = vector(#vn, k, sigma(vn[k])); svn = Set(vn); if (#select(x->x==n, svn), knb = nb); if (!#select(x->x<=n, svn), ok = 0);); knb;}

PARI

apply( A286011(n)=if(n<3,2-n, n=invsigma(n), vecmax(apply(self,n))+1), [1..99]) \\ See Alekseyev-link for invsigma(). - M. F. Hasler, Nov 20 2019