login
Numbers that require a record number of iterations of the sum of odd divisors function (A000593) to reach 1.
0

%I #41 Oct 30 2018 04:26:06

%S 1,2,3,5,9,17,67,193,1069,2137,4273,34183,205097,990361,11884331,

%T 38294881,76589761,574396453,10339136153,36177024721,72354049441,

%U 144708098881

%N Numbers that require a record number of iterations of the sum of odd divisors function (A000593) to reach 1.

%C It seems that 9 is the only composite term.

%C a(2)-a(14) appear in De Koninck's book.

%C a(2)-a(18) were calculated by Kim & Bayad.

%D J.-M. De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 54, entry 193.

%H Daeyeoul Kim and Abdelmejid Bayad, <a href="https://doi.org/10.1080/10586458.2016.1162231">Polygon Numbers Associated with the Sum of Odd Divisors Function</a>, Experimental Mathematics, Vol. 26, No. 3 (2017), pp. 287-297.

%e 5 is in the sequence since iterating A000593 on 5, i.e. A000593(5) = 6, A000593(6) = 4, A000593(4) = 1, reaches 1 after 3 steps, more steps than for any number below 5.

%t oddsigma[n_] := If[ n < 1, 0, Times @@ (If[ # < 3, 1, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)]; niter[n_] := Module[{c=0}, m=n; While[m>1, m = oddsigma[m]; c++]; c]; seq={}; sm=-1; Do[s=niter[n]; If[s>sm, AppendTo[seq, n]; sm=s],{n,1,10000}]; seq (* after Michael Somos at A000593 *)

%Y Cf. A000593.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Oct 29 2018

%E a(19)-a(22) from _Giovanni Resta_, Oct 29 2018