OFFSET
1,2
COMMENTS
Also called log-concave-down 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 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
12: {1,1,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
36: {1,1,2,2}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
52: {1,1,6}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
66: {1,2,5}
68: {1,1,7}
72: {1,1,1,2,2}
76: {1,1,8}
78: {1,2,6}
80: {1,1,1,1,3}
84: {1,1,2,4}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], GreaterEqual@@Divide@@@Reverse/@Partition[primeptn[#], 2, 1]&]
CROSSREFS
The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A242031.
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A000929 counts partitions with all adjacent parts x >= 2y.
A002843 counts compositions with all adjacent parts x <= 2y.
A167865 counts strict chains of divisors > 1 summing to n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 23 2021
STATUS
approved