

A342496


Number of integer partitions of n with constant (equal) first quotients.


8



1, 1, 2, 3, 4, 4, 6, 6, 7, 7, 8, 7, 11, 9, 11, 12, 12, 10, 14, 12, 15, 16, 14, 13, 19, 15, 17, 17, 20, 16, 23, 19, 21, 20, 20, 22, 26, 21, 23, 25, 28, 22, 30, 24, 27, 29, 26, 25, 33, 29, 30, 29, 32, 28, 34, 31, 36, 34, 32, 31, 42
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OFFSET

0,3


COMMENTS

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).


LINKS

Table of n, a(n) for n=0..60.
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.


FORMULA

a(n > 0) = (A342495(n) + A000005(n))/2.


EXAMPLE

The partition (12,6,3) has first quotients (1/2,1/2) so is counted under a(21).
The a(1) = 1 through a(9) = 7 partitions:
1 2 3 4 5 6 7 8 9
11 21 22 32 33 43 44 54
111 31 41 42 52 53 63
1111 11111 51 61 62 72
222 421 71 81
111111 1111111 2222 333
11111111 111111111


MATHEMATICA

Table[Length[Select[IntegerPartitions[n], SameQ@@Divide@@@Partition[#, 2, 1]&]], {n, 0, 30}]


CROSSREFS

The version for differences instead of quotients is A049988.
The ordered version is A342495.
The distinct version is A342514.
The strict case is A342515.
The Heinz numbers of these partitions are A342522.
A000005 counts constant partitions.
A003238 counts chains of divisors summing to n  1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A000837, A002843, A003242, A074206, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
Sequence in context: A307989 A160680 A243069 * A061984 A337125 A063208
Adjacent sequences: A342493 A342494 A342495 * A342497 A342498 A342499


KEYWORD

nonn


AUTHOR

Gus Wiseman, Mar 17 2021


STATUS

approved



