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A342530
Number of strict chains of divisors ending with n and having distinct first quotients.
6
1, 2, 2, 3, 2, 6, 2, 6, 3, 6, 2, 12, 2, 6, 6, 9, 2, 12, 2, 12, 6, 6, 2, 28, 3, 6, 6, 12, 2, 26, 2, 14, 6, 6, 6, 31, 2, 6, 6, 28, 2, 26, 2, 12, 12, 6, 2, 52, 3, 12, 6, 12, 2, 28, 6, 28, 6, 6, 2, 66, 2, 6, 12, 25, 6, 26, 2, 12, 6, 26, 2, 76, 2, 6, 12, 12, 6, 26
OFFSET
1,2
COMMENTS
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the quotients of (6,3,1) are (1/2,1/3).
EXAMPLE
The a(1) = 1 through a(12) = 12 chains (reversed):
1 2 3 4 5 6 7 8 9 10 11 12
2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 10/1 11/1 12/1
4/2 6/2 8/2 9/3 10/2 12/2
6/3 8/4 10/5 12/3
6/2/1 8/2/1 10/2/1 12/4
6/3/1 8/4/1 10/5/1 12/6
12/2/1
12/3/1
12/4/1
12/4/2
12/6/1
12/6/2
Not counted under a(12) are: 12/4/2/1, 12/6/2/1, 12/6/3, 12/6/3/1.
MATHEMATICA
cmi[n_]:=Prepend[Prepend[#, n]&/@Join@@cmi/@Most[Divisors[n]], {n}];
Table[Length[Select[cmi[n], UnsameQ@@Divide@@@Partition[#, 2, 1]&]], {n, 100}]
CROSSREFS
The version for weakly increasing first quotients is A057567.
The version for equal first quotients is A169594.
The case of chains starting with 1 is A254578.
The version for strictly increasing first quotients is A342086.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A067824 counts strict chains of divisors ending with n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A334997 counts chains of divisors of n by length.
A342495/A342529 count compositions with equal/distinct quotients.
A342496/A342514 count partitions with equal/distinct quotients.
A342515/A342520 count strict partitions with equal/distinct quotients.
A342522/A342521 rank partitions with equal/distinct quotients.
Sequence in context: A197929 A326849 A328706 * A240090 A078224 A159688
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2021
STATUS
approved