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

1,4

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

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

Example

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.

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

Prog

(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

Crossrefs

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

Sequence in context: A368514 A289229 A263097 * A241954 A307047 A049600

Adjacent sequences: A286008 A286009 A286010 * A286012 A286013 A286014

Keyword

nonn

Author

Michel Marcus, Apr 30 2017

Status

approved