A342523 - OEIS

%I #8 Mar 23 2021 16:10:36

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,

%T 28,29,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,53,

%U 55,56,57,58,59,61,62,63,64,65,66,67,68,69,71,73,74,76

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

%C Also called log-concave-up 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 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 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    18: {1,2,2}

%e    30: {1,2,3}

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

%e    50: {1,3,3}

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

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

%e    70: {1,3,4}

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

%e    75: {2,3,3}

%e    90: {1,2,2,3}

%e    98: {1,4,4}

%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],LessEqual@@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 A304678.

%Y For differences instead of quotients we have A325360 (count: A240026).

%Y These partitions are counted by A342523 (strict: A342516, ordered: A342492).

%Y The strictly increasing version is A342524.

%Y The weakly decreasing version is A342526.

%Y A000041 counts partitions (strict: A000009).

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

%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 A342086 counts strict chains of divisors with strictly increasing quotients.

%Y Cf. A000005, A002843, A056239, A067824, A112798, A124010, A130091, A238710, A253249, A325351, A325352, A342191.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 23 2021