%I #7 Mar 23 2021 16:10:43

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,19,20,21,22,23,25,26,28,29,31,

%T 33,34,35,37,38,39,41,43,44,45,46,47,49,51,52,53,55,57,58,59,61,62,63,

%U 65,66,67,68,69,71,73,74,76,77,78,79,82,83,85,86,87,89,91

%N Heinz numbers of integer partitions with strictly increasing first quotients.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%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="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>

%e The prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.

%e Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:

%e 8: {1,1,1}

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

%e 18: {1,2,2}

%e 24: {1,1,1,2}

%e 27: {2,2,2}

%e 30: {1,2,3}

%e 32: {1,1,1,1,1}

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

%e 42: {1,2,4}

%e 48: {1,1,1,1,2}

%e 50: {1,3,3}

%e 54: {1,2,2,2}

%e 56: {1,1,1,4}

%e 60: {1,1,2,3}

%e 64: {1,1,1,1,1,1}

%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[100],Less@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

%Y For differences instead of quotients we have A325456 (count: A240027).

%Y For multiplicities (prime signature) instead of quotients we have A334965.

%Y The version counting strict divisor chains is A342086.

%Y These partitions are counted by A342498 (strict: A342517, ordered: A342493).

%Y The weakly increasing version is A342523.

%Y The strictly decreasing version is A342525.

%Y A001055 counts factorizations (strict: A045778, ordered: A074206).

%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).

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

%Y A318991/A318992 rank reversed partitions with/without integer quotients.

%Y A342098 counts (strict) partitions with all adjacent parts x > 2y.

%Y Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 23 2021