silly but fun (recreational purposes only)

[wouter meeussen]

being in a festive mood, (and up to no good):

what is the sequence that remains unchanged under the operation: "replace every integer k by the
(sorted) sequence of divisors of k+1"? What is the length of this list at step n?

And, no, I don't intend to contaminate\decorate the good OEIS with such 'festoons' unless a GF or
'property' were connected to it. We can have some fun too, can't we Neil?

W.
(spoiler below)
















































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invariant of Flatten[# /. k_ :>Divisors[1+k] ]

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


length of list after n steps: (ziltch from SuperSeeker)
{1, 2, 4, 9, 19, 43, 94, 210, 464, 1035, 2295, 5111, 11352,
25259, 56145, 124888, 277669}

it even produces a triangular table if you're in the mood for digit-counting: how many ones, two's,
three's .. at each step?
{1},
{1,1},
{2,1,1},
{4,3,1,1},
{9,5,3,1,1},
{19,13,6,3,1,1},
{43,26,14,6,3,1,1},
{94,61,29,15,6,3,1,1},
{210,130,68,30,15,6,3,1,1},
{464,297,146,71,31,15,6,3,1,1},
{1035,648,331,152,72,31,15,6,3,1,1},
{2295,1457,727,347,155,73,31,15,6,3,1,1},
{5111,3215,1628,759,353,156,73,31,15,6,3,1,1},
{11352,7184,3602,1704,774,356,157,73,31,15,6,3,1,1},
{25259,15923,8038,3765,1736,780,357,157,73,31,15,6,3,1,1},
{56145,35482,17832,8411,3839,1751,783,358,157,73,31,15,6,3,1,1},
{124888,78794,39713,18647,8571,3870,1757,784,358,157,73,31,15,6,3,1,1}

with T[n+1,1]= row-sum of n-th row (of course).
Tail end backwards: 1,1,3,6,15,31,73,157,358,.. does SuperSeek something,
and had me screaming for help earlier-on.
Would it then be worth its weight in bits after all?

Nah!
more terms show that it's nothing. I vote that we forget about it!

a dirty program for row 16 of the triangle:
Nest[Apply[Plus,Map[Last,Split[Sort[Apply[Sequence,Thread[w[Divisors[1+Range[Length[#]]]& @
#,List/@#]]/. w->(Outer[Sequence,##]&),{1}]],First[#1]===First[#2]&],{2}],{1}]&,{1},16]


W.