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A342522 Heinz numbers of integer partitions with constant (equal) first quotients.

%I A342522 #9 Feb 16 2025 08:34:01
%S A342522 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,31,32,
%T A342522 33,34,35,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,65,67,
%U A342522 69,71,73,74,77,79,81,82,83,85,86,87,89,91,93,94,95,97
%N A342522 Heinz numbers of integer partitions with constant (equal) first quotients.
%C A342522 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 A342522 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 A342522 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342522 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%H A342522 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients.</a>
%e A342522 The prime indices of 2093 are {4,6,9}, with first quotients (3/2,3/2), so 2093 is in the sequence.
%e A342522 Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
%e A342522    12: {1,1,2}
%e A342522    18: {1,2,2}
%e A342522    20: {1,1,3}
%e A342522    24: {1,1,1,2}
%e A342522    28: {1,1,4}
%e A342522    30: {1,2,3}
%e A342522    36: {1,1,2,2}
%e A342522    40: {1,1,1,3}
%e A342522    44: {1,1,5}
%e A342522    45: {2,2,3}
%e A342522    48: {1,1,1,1,2}
%e A342522    50: {1,3,3}
%e A342522    52: {1,1,6}
%e A342522    54: {1,2,2,2}
%e A342522    56: {1,1,1,4}
%e A342522    60: {1,1,2,3}
%e A342522    63: {2,2,4}
%e A342522    66: {1,2,5}
%t A342522 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A342522 Select[Range[100],SameQ@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
%Y A342522 For multiplicities (prime signature) instead of quotients we have A072774.
%Y A342522 The version counting strict divisor chains is A169594.
%Y A342522 For differences instead of quotients we have A325328 (count: A049988).
%Y A342522 These partitions are counted by A342496 (strict: A342515, ordered: A342495).
%Y A342522 The distinct instead of equal version is A342521.
%Y A342522 A000005 count constant partitions.
%Y A342522 A000041 counts partitions (strict: A000009).
%Y A342522 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342522 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342522 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342522 A318991/A318992 rank reversed partitions with/without integer quotients.
%Y A342522 A342086 counts strict chains of divisors with strictly increasing quotients.
%Y A342522 Cf. A003242, A047966, A049980, A056239, A067824, A112798, A118914, A124010, A130091, A181819, A325351, A325352.
%K A342522 nonn
%O A342522 1,2
%A A342522 _Gus Wiseman_, Mar 23 2021