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A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.

%I A342525 #8 Feb 16 2025 08:34:01
%S A342525 1,2,3,4,5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,29,30,31,33,
%T A342525 34,35,37,38,39,41,43,46,47,49,50,51,53,55,57,58,59,61,62,65,67,69,70,
%U A342525 71,73,74,75,77,79,82,83,85,86,87,89,91,93,94,95,97,98
%N A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.
%C A342525 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 A342525 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 A342525 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342525 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%H A342525 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>
%e A342525 The prime indices of 150 are {1,2,3,3}, with first quotients (2,3/2,1), so 150 is in the sequence.
%e A342525 Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
%e A342525     8: {1,1,1}
%e A342525    12: {1,1,2}
%e A342525    16: {1,1,1,1}
%e A342525    20: {1,1,3}
%e A342525    24: {1,1,1,2}
%e A342525    27: {2,2,2}
%e A342525    28: {1,1,4}
%e A342525    32: {1,1,1,1,1}
%e A342525    36: {1,1,2,2}
%e A342525    40: {1,1,1,3}
%e A342525    42: {1,2,4}
%e A342525    44: {1,1,5}
%e A342525    45: {2,2,3}
%e A342525    48: {1,1,1,1,2}
%t A342525 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A342525 Select[Range[100],Greater@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
%Y A342525 For multiplicities (prime signature) instead of quotients we have A304686.
%Y A342525 For differences instead of quotients we have A325457 (count: A320470).
%Y A342525 The version counting strict divisor chains is A342086.
%Y A342525 These partitions are counted by A342499 (strict: A342518, ordered: A342494).
%Y A342525 The strictly increasing version is A342524.
%Y A342525 The weakly decreasing version is A342526.
%Y A342525 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342525 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342525 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342525 A318991/A318992 rank reversed partitions with/without integer quotients.
%Y A342525 A342098 counts (strict) partitions with all adjacent parts x > 2y.
%Y A342525 Cf. A056239, A067824, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A325405, A334997, A342530.
%K A342525 nonn
%O A342525 1,2
%A A342525 _Gus Wiseman_, Mar 23 2021