OFFSET
1,2
COMMENTS
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 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
8: {1,1,1}
16: {1,1,1,1}
18: {1,2,2}
24: {1,1,1,2}
27: {2,2,2}
30: {1,2,3}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
48: {1,1,1,1,2}
50: {1,3,3}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], Less@@Divide@@@Reverse/@Partition[primeptn[#], 2, 1]&]
CROSSREFS
For multiplicities (prime signature) instead of quotients we have A334965.
The version counting strict divisor chains is A342086.
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A167865 counts strict chains of divisors > 1 summing to n.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 23 2021
STATUS
approved