A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

1, 2, 4, 2, 5, 1, 5, 2, 7, 4, 15, 3, 13, 3, 2, 2, 13, 4, 12, 5, 2, 13, 16, 2, 17, 4, 9, 1, 78, 7, 10, 4, 17, 11, 6, 5, 28, 22, 4, 7, 39, 2, 16, 16, 16, 10, 32, 5, 13, 17, 9, 3, 58, 11, 19, 5, 13, 67, 97, 2, 23, 5, 16, 2, 4, 8, 101, 21, 19, 11, 50, 4, 20, 20, 23, 14, 21, 10, 36, 5, 15

Offset

1,2

Comments

Let sigma^m(n) be result of applying sum-of-divisors function m times to n; sequence gives m(n) = min m such that n divides sigma^m(n).

Perfect numbers require one iteration.

It is conjectured that the sequence is finite for all n.

See also the Cohen-te Riele links under A019276.

a(A111227(n)) > A111227(n). - Reinhard Zumkeller, Aug 02 2012

a(659) > 870. - Michel Marcus, Jan 04 2017

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

Links

Don Reble, Table of n, a(n) for n = 1..1578 (Terms a(1..400) from T. D. Noe, Nov 2007; a(401..659) from Michel Marcus, Jan 02 2017), Feb 20 2022.

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100. See Table 2, p. 95.

Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.

Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

Carl Pomerance, On the composition of the functions sigma and phi, Colloq. Math., 59 (1989), 11-15.

Wikipedia, Iterated function, as of Jan 02 2020.

Zeraoulia Rafik, On congruence of the iterated form sigma^k(m) = 0 mod m, arXiv:2102.09941 [math.NT], 2021.

Formula

Conjecture: lim_{n -> oo} log(Sum_{k=1..n} a(k))/log(n) = C = 1.6... - Benoit Cloitre, Aug 24 2002

From Michel Marcus, Jan 02 2017: (Start)

a(n) = 1 for n in A007691.

a(n) = 2 for n in A019278 unless it belongs to A007691.

a(n) = 3 for n in A019292 unless it belongs to A007691 or A019278. (End)

Example

If n = 9 the iteration sequence is s(9) = {9, 13, 14, 24, 60, 168, 480, 1512, 4800, 15748, 28672} and Mod[s(9), 9] = {0, 4, 5, 6, 6, 6, 3, 0, 3, 7, 7}. The first iterate which is a multiple of 9 is the 7th = 1512, so a(9) = 7. For n = 67, the 101st iterate is the first, so a(67) = 101. Usually several iterates are divisible by the initial value. E.g., if n = 6, then 91 of the first 100 iterates are divisible by 6.
A difficult term to compute: a(461) = 557. - Don Reble, Jun 23 2005

Maple

A019294 := proc(n)
    local a, nitr ;
    a := 1 ;
    nitr := numtheory[sigma](n);
    while modp(nitr, n) <> 0 do
        nitr := numtheory[sigma](nitr) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 22 2016

Mathematica

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[ Mod[d, m] == 0, 0, d]]; Table[ Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1, {n, 84}] (* Robert G. Wilson v, Jun 24 2005 *)
Table[Length[NestWhileList[DivisorSigma[1, #]&, DivisorSigma[1, n], !Divisible[ #, n]&]], {n, 90}] (* Harvey P. Dale, Mar 04 2015 *)

Prog

(PARI) a(n)=if(n<0, 0, c=1; s=n; while(sigma(s)%n>0, s=sigma(s); c++); c)
(PARI) apply( A019294(n, s=n)=for(k=1, oo, (s=sigma(s))%n||return(k)), [1..99]) \\ M. F. Hasler, Jan 07 2020
(Haskell)
a019294 n = snd $ until ((== 0) . (`mod` n) . fst)
                        (\(x, i) -> (a000203 x, i + 1)) (a000203 n, 1)
-- Reinhard Zumkeller, Aug 02 2012
(Magma) a:=[]; f:=func<n|DivisorSigma(1, n)>; for n in [1..81] do k:=n; s:=1; while f(k) mod n ne 0 do k:=f(k); s:=s+1; end while; Append(~a, s); end for; a; // Marius A. Burtea, Jan 11 2020

Crossrefs

Cf. A019295 (ratio sigma^m(n)/n), A019276 (indices of records), A019277 (records), A000396.

Sequence in context: A263424 A279527 A347184 * A238262 A307664 A057037

Adjacent sequences: A019291 A019292 A019293 * A019295 A019296 A019297

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Additional comments from Labos Elemer, Jun 20 2001

Edited by M. F. Hasler, Jan 07 2020

Status

approved