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A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

%I A342526 #8 Feb 16 2025 08:34:01
%S A342526 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 A342526 31,32,33,34,35,37,38,39,41,42,43,46,47,49,50,51,53,54,55,57,58,59,61,
%U A342526 62,64,65,67,69,70,71,73,74,75,77,79,81,82,83,85,86,87
%N A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.
%C A342526 Also called log-concave-down partitions.
%C A342526 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 A342526 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 A342526 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342526 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%H A342526 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>
%e A342526 The prime indices of 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
%e A342526 Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
%e A342526    12: {1,1,2}
%e A342526    20: {1,1,3}
%e A342526    24: {1,1,1,2}
%e A342526    28: {1,1,4}
%e A342526    36: {1,1,2,2}
%e A342526    40: {1,1,1,3}
%e A342526    44: {1,1,5}
%e A342526    45: {2,2,3}
%e A342526    48: {1,1,1,1,2}
%e A342526    52: {1,1,6}
%e A342526    56: {1,1,1,4}
%e A342526    60: {1,1,2,3}
%e A342526    63: {2,2,4}
%e A342526    66: {1,2,5}
%e A342526    68: {1,1,7}
%e A342526    72: {1,1,1,2,2}
%e A342526    76: {1,1,8}
%e A342526    78: {1,2,6}
%e A342526    80: {1,1,1,1,3}
%e A342526    84: {1,1,2,4}
%t A342526 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A342526 Select[Range[100],GreaterEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
%Y A342526 The version counting strict divisor chains is A057567.
%Y A342526 For multiplicities (prime signature) instead of quotients we have A242031.
%Y A342526 For differences instead of quotients we have A325361 (count: A320466).
%Y A342526 These partitions are counted by A342513 (strict: A342519, ordered: A069916).
%Y A342526 The weakly increasing version is A342523.
%Y A342526 The strictly decreasing version is A342525.
%Y A342526 A000929 counts partitions with all adjacent parts x >= 2y.
%Y A342526 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342526 A002843 counts compositions with all adjacent parts x <= 2y.
%Y A342526 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342526 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342526 A318991/A318992 rank reversed partitions with/without integer quotients.
%Y A342526 Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.
%K A342526 nonn
%O A342526 1,2
%A A342526 _Gus Wiseman_, Mar 23 2021