%I #7 Mar 23 2021 16:10:57

%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,25,26,27,29,30,

%T 31,32,33,34,35,37,38,39,41,42,43,46,47,49,50,51,53,54,55,57,58,59,61,

%U 62,64,65,67,69,70,71,73,74,75,77,79,81,82,83,85,86,87

%N Heinz numbers of integer partitions with weakly decreasing first quotients.

%C Also called log-concave-down partitions.

%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 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.

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

%e 12: {1,1,2}

%e 20: {1,1,3}

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

%e 28: {1,1,4}

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

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

%e 44: {1,1,5}

%e 45: {2,2,3}

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

%e 52: {1,1,6}

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

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

%e 63: {2,2,4}

%e 66: {1,2,5}

%e 68: {1,1,7}

%e 72: {1,1,1,2,2}

%e 76: {1,1,8}

%e 78: {1,2,6}

%e 80: {1,1,1,1,3}

%e 84: {1,1,2,4}

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

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

%Y The version counting strict divisor chains is A057567.

%Y For multiplicities (prime signature) instead of quotients we have A242031.

%Y For differences instead of quotients we have A325361 (count: A320466).

%Y These partitions are counted by A342513 (strict: A342519, ordered: A069916).

%Y The weakly increasing version is A342523.

%Y The strictly decreasing version is A342525.

%Y A000929 counts partitions with all adjacent parts x >= 2y.

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

%Y A002843 counts compositions with all adjacent parts x <= 2y.

%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 Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 23 2021