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A342496 Number of integer partitions of n with constant (equal) first quotients.

%I A342496 #16 Feb 16 2025 08:34:01
%S A342496 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 A342496 17,20,16,23,19,21,20,20,22,26,21,23,25,28,22,30,24,27,29,26,25,33,29,
%U A342496 30,29,32,28,34,31,36,34,32,31,42
%N A342496 Number of integer partitions of n with constant (equal) first quotients.
%C A342496 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 A342496 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342496 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%F A342496 a(n > 0) = (A342495(n) + A000005(n))/2.
%e A342496 The partition (12,6,3) has first quotients (1/2,1/2) so is counted under a(21).
%e A342496 The a(1) = 1 through a(9) = 7 partitions:
%e A342496   1   2    3     4      5       6        7         8          9
%e A342496       11   21    22     32      33       43        44         54
%e A342496            111   31     41      42       52        53         63
%e A342496                  1111   11111   51       61        62         72
%e A342496                                 222      421       71         81
%e A342496                                 111111   1111111   2222       333
%e A342496                                                    11111111   111111111
%t A342496 Table[Length[Select[IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]
%Y A342496 The version for differences instead of quotients is A049988.
%Y A342496 The ordered version is A342495.
%Y A342496 The distinct version is A342514.
%Y A342496 The strict case is A342515.
%Y A342496 The Heinz numbers of these partitions are A342522.
%Y A342496 A000005 counts constant partitions.
%Y A342496 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342496 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342496 Cf. A000837, A002843, A003242, A074206, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K A342496 nonn
%O A342496 0,3
%A A342496 _Gus Wiseman_, Mar 17 2021