cp's OEIS Frontend

A342521 Heinz numbers of integer partitions with distinct first quotients.

%I A342521 #9 Feb 16 2025 08:34:01
%S A342521 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 A342521 30,31,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,
%U A342521 60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82
%N A342521 Heinz numbers of integer partitions with distinct first quotients.
%C A342521 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 A342521 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 A342521 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342521 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>
%e A342521 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 A342521 Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
%e A342521     8: {1,1,1}
%e A342521    16: {1,1,1,1}
%e A342521    24: {1,1,1,2}
%e A342521    27: {2,2,2}
%e A342521    32: {1,1,1,1,1}
%e A342521    36: {1,1,2,2}
%e A342521    40: {1,1,1,3}
%e A342521    42: {1,2,4}
%e A342521    48: {1,1,1,1,2}
%e A342521    54: {1,2,2,2}
%e A342521    56: {1,1,1,4}
%e A342521    64: {1,1,1,1,1,1}
%e A342521    72: {1,1,1,2,2}
%e A342521    80: {1,1,1,1,3}
%e A342521    81: {2,2,2,2}
%e A342521    84: {1,1,2,4}
%e A342521    88: {1,1,1,5}
%e A342521    96: {1,1,1,1,1,2}
%e A342521   100: {1,1,3,3}
%t A342521 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A342521 Select[Range[100],UnsameQ@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
%Y A342521 For multiplicities (prime signature) instead of quotients we have A130091.
%Y A342521 For differences instead of quotients we have A325368 (count: A325325).
%Y A342521 These partitions are counted by A342514 (strict: A342520, ordered: A342529).
%Y A342521 The equal instead of distinct version is A342522.
%Y A342521 The version counting strict divisor chains is A342530.
%Y A342521 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342521 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342521 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342521 A318991/A318992 rank reversed partitions with/without integer quotients.
%Y A342521 Cf. A003242, A005117, A056239, A067824, A098859, A112798, A169594, A253249, A325326, A325337, A325405.
%K A342521 nonn
%O A342521 1,2
%A A342521 _Gus Wiseman_, Mar 23 2021