A342521 - OEIS

%I #8 Mar 23 2021 16:10:17

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,28,29,

%T 30,31,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,

%U 60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82

%N Heinz numbers of integer partitions with distinct 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="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>

%e The prime indices of 1365 are {2,3,4,6}, with first quotients (3/2,4/3,3/2), so 1365 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    24: {1,1,1,2}

%e    27: {2,2,2}

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

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

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

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

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

%e    81: {2,2,2,2}

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

%e    88: {1,1,1,5}

%e    96: {1,1,1,1,1,2}

%e   100: {1,1,3,3}

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

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

%Y For multiplicities (prime signature) instead of quotients we have A130091.

%Y For differences instead of quotients we have A325368 (count: A325325).

%Y These partitions are counted by A342514 (strict: A342520, ordered: A342529).

%Y The equal instead of distinct version is A342522.

%Y The version counting strict divisor chains is A342530.

%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 Cf. A003242, A005117, A056239, A067824, A098859, A112798, A169594, A253249, A325326, A325337, A325405.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 23 2021