A320641 Numbers that require a record number of iterations of the sum of odd divisors function (A000593) to reach 1.

1, 2, 3, 5, 9, 17, 67, 193, 1069, 2137, 4273, 34183, 205097, 990361, 11884331, 38294881, 76589761, 574396453, 10339136153, 36177024721, 72354049441, 144708098881

Offset

1,2

Comments

It seems that 9 is the only composite term.

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

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

References

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

Links

Table of n, a(n) for n=1..22.

Daeyeoul Kim and Abdelmejid Bayad, Polygon Numbers Associated with the Sum of Odd Divisors Function, Experimental Mathematics, Vol. 26, No. 3 (2017), pp. 287-297.

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A000593.

Sequence in context: A014227 A334816 A064769 * A365640 A047021 A201359

Adjacent sequences: A320638 A320639 A320640 * A320642 A320643 A320644

Keyword

nonn,more

Author

Amiram Eldar, Oct 29 2018

Extensions

a(19)-a(22) from Giovanni Resta, Oct 29 2018

Status

approved