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Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.
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%I #11 Aug 25 2020 17:16:28

%S 1,1,1,1,1,1,2,1,1,1,1,3,2,1,1,1,3,3,1,1,2,1,1,3,2,1,1,1,5,7,3,1,1,1,

%T 3,2,1,3,2,1,4,6,4,1,1,1,1,5,7,3,1,1,1,5,7,3,1,3,2,1,3,2,1,1,1,7,15,

%U 13,4,1,2,1,1,3,2,1,3,3,1,1,5,7,3,1,1,1

%N Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.

%H Alois P. Heinz, <a href="/A337255/b337255.txt">Rows n = 1..5000, flattened</a>

%e Sequence of rows begins:

%e 1: {1} 16: {1,4,6,4,1}

%e 2: {1,1} 17: {1,1}

%e 3: {1,1} 18: {1,5,7,3}

%e 4: {1,2,1} 19: {1,1}

%e 5: {1,1} 20: {1,5,7,3}

%e 6: {1,3,2} 21: {1,3,2}

%e 7: {1,1} 22: {1,3,2}

%e 8: {1,3,3,1} 23: {1,1}

%e 9: {1,2,1} 24: {1,7,15,13,4}

%e 10: {1,3,2} 25: {1,2,1}

%e 11: {1,1} 26: {1,3,2}

%e 12: {1,5,7,3} 27: {1,3,3,1}

%e 13: {1,1} 28: {1,5,7,3}

%e 14: {1,3,2} 29: {1,1}

%e 15: {1,3,2} 30: {1,7,12,6}

%e Row n = 24 counts the following chains:

%e 24 24/1 24/2/1 24/4/2/1 24/8/4/2/1

%e 24/2 24/3/1 24/6/2/1 24/12/4/2/1

%e 24/3 24/4/1 24/6/3/1 24/12/6/2/1

%e 24/4 24/4/2 24/8/2/1 24/12/6/3/1

%e 24/6 24/6/1 24/8/4/1

%e 24/8 24/6/2 24/8/4/2

%e 24/12 24/6/3 24/12/2/1

%e 24/8/1 24/12/3/1

%e 24/8/2 24/12/4/1

%e 24/8/4 24/12/4/2

%e 24/12/1 24/12/6/1

%e 24/12/2 24/12/6/2

%e 24/12/3 24/12/6/3

%e 24/12/4

%e 24/12/6

%p b:= proc(n) option remember; expand(x*(1 +

%p add(b(d), d=numtheory[divisors](n) minus {n})))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):

%p seq(T(n), n=1..50); # _Alois P. Heinz_, Aug 23 2020

%t chss[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chss[d],{d,Most[Divisors[n]]}],{n}];

%t Table[Length[Select[chss[n],Length[#]==k&]],{n,30},{k,1+PrimeOmega[n]}]

%Y A008480 gives rows ends.

%Y A067824 gives row sums.

%Y A073093 gives row lengths.

%Y A334996 appears to be the case of chains ending with 1.

%Y A337071 is the sum of row n!.

%Y A000005 counts divisors.

%Y A001055 counts factorizations.

%Y A001222 counts prime factors with multiplicity.

%Y A067824 counts chains of divisors starting with n.

%Y A074206 counts chains of divisors from n to 1.

%Y A122651 counts chains of divisors summing to n.

%Y A167865 counts chains of divisors > 1 summing to n.

%Y A251683 counts chains of divisors from n to 1 by length.

%Y A253249 counts nonempty chains of divisors.

%Y A337070 counts chains of divisors starting with A006939(n).

%Y A337256 counts chains of divisors.

%Y Cf. A001221, A002033, A124010, A124433, A337074, A337105, A337107.

%K nonn,tabf

%O 1,7

%A _Gus Wiseman_, Aug 23 2020