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A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

%I A342524 #8 Feb 16 2025 08:34:01
%S A342524 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 A342524 33,34,35,37,38,39,41,43,44,45,46,47,49,51,52,53,55,57,58,59,61,62,63,
%U A342524 65,66,67,68,69,71,73,74,76,77,78,79,82,83,85,86,87,89,91
%N A342524 Heinz numbers of integer partitions with strictly increasing first quotients.
%C A342524 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 A342524 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 A342524 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342524 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%H A342524 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>
%e A342524 The prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
%e A342524 Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
%e A342524     8: {1,1,1}
%e A342524    16: {1,1,1,1}
%e A342524    18: {1,2,2}
%e A342524    24: {1,1,1,2}
%e A342524    27: {2,2,2}
%e A342524    30: {1,2,3}
%e A342524    32: {1,1,1,1,1}
%e A342524    36: {1,1,2,2}
%e A342524    40: {1,1,1,3}
%e A342524    42: {1,2,4}
%e A342524    48: {1,1,1,1,2}
%e A342524    50: {1,3,3}
%e A342524    54: {1,2,2,2}
%e A342524    56: {1,1,1,4}
%e A342524    60: {1,1,2,3}
%e A342524    64: {1,1,1,1,1,1}
%t A342524 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A342524 Select[Range[100],Less@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
%Y A342524 For differences instead of quotients we have A325456 (count: A240027).
%Y A342524 For multiplicities (prime signature) instead of quotients we have A334965.
%Y A342524 The version counting strict divisor chains is A342086.
%Y A342524 These partitions are counted by A342498 (strict: A342517, ordered: A342493).
%Y A342524 The weakly increasing version is A342523.
%Y A342524 The strictly decreasing version is A342525.
%Y A342524 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342524 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342524 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342524 A318991/A318992 rank reversed partitions with/without integer quotients.
%Y A342524 A342098 counts (strict) partitions with all adjacent parts x > 2y.
%Y A342524 Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.
%K A342524 nonn
%O A342524 1,2
%A A342524 _Gus Wiseman_, Mar 23 2021