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.
Wikipedia, Arithmetic progression
EXAMPLE
The prime indices of 2093 are {4,6,9}, with first quotients (3/2,3/2), so 2093 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}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
50: {1,3,3}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
66: {1,2,5}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], SameQ@@Divide@@@Reverse/@Partition[primeptn[#], 2, 1]&]
CROSSREFS
For multiplicities (prime signature) instead of quotients we have A072774.
The version counting strict divisor chains is A169594.
The distinct instead of equal version is A342521.
A000005 count constant partitions.
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