OFFSET
1,2
COMMENTS
Also called log-concave-up partitions.
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.
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).
LINKS
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
EXAMPLE
The prime indices of 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
18: {1,2,2}
30: {1,2,3}
36: {1,1,2,2}
50: {1,3,3}
54: {1,2,2,2}
60: {1,1,2,3}
70: {1,3,4}
72: {1,1,1,2,2}
75: {2,3,3}
90: {1,2,2,3}
98: {1,4,4}
100: {1,1,3,3}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], LessEqual@@Divide@@@Reverse/@Partition[primeptn[#], 2, 1]&]
CROSSREFS
The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A304678.
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A000929 counts partitions with adjacent parts x >= 2y.
A167865 counts strict chains of divisors > 1 summing to n.
A342086 counts strict chains of divisors with strictly increasing quotients.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 23 2021
STATUS
approved