OFFSET
1,1
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 term rucksack is short for run-knapsack.
EXAMPLE
The terms together with their prime indices begin:
12: {1,1,2}
24: {1,1,1,2}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
60: {1,1,2,3}
63: {2,2,4}
72: {1,1,1,2,2}
80: {1,1,1,1,3}
84: {1,1,2,4}
96: {1,1,1,1,1,2}
108: {1,1,2,2,2}
112: {1,1,1,1,4}
120: {1,1,1,2,3}
126: {1,2,2,4}
132: {1,1,2,5}
144: {1,1,1,1,2,2}
156: {1,1,2,6}
160: {1,1,1,1,1,3}
168: {1,1,1,2,4}
For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !UnsameQ@@Total/@primeMS/@Select[Divisors[#], PrimePowerQ]&]
CROSSREFS
Non-knapsack partitions are ranked by A299729.
The complement for compositions is counted by A354580.
A073093 counts prime-power divisors.
A353861 counts distinct partial run-sums of prime indices.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 15 2022
STATUS
approved