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%I #15 Feb 21 2023 13:23:34
%S 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,
%T 17,20,16,23,19,21,20,20,22,26,21,23,25,28,22,30,24,27,29,26,25,33,29,
%U 30,29,32,28,34,31,36,34,32,31,42
%N Number of integer partitions of n with constant (equal) first quotients.
%C 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).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%F a(n > 0) = (A342495(n) + A000005(n))/2.
%e The partition (12,6,3) has first quotients (1/2,1/2) so is counted under a(21).
%e The a(1) = 1 through a(9) = 7 partitions:
%e 1 2 3 4 5 6 7 8 9
%e 11 21 22 32 33 43 44 54
%e 111 31 41 42 52 53 63
%e 1111 11111 51 61 62 72
%e 222 421 71 81
%e 111111 1111111 2222 333
%e 11111111 111111111
%t Table[Length[Select[IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]
%Y The version for differences instead of quotients is A049988.
%Y The ordered version is A342495.
%Y The distinct version is A342514.
%Y The strict case is A342515.
%Y The Heinz numbers of these partitions are A342522.
%Y A000005 counts constant partitions.
%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y Cf. A000837, A002843, A003242, A074206, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 17 2021