A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

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.

In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short.

Links

Table of n, a(n) for n=1..67.

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
The sequence contains 18 because its prime-power divisors {1,2,3,9} have prime indices {}, {1}, {2}, {2,2} with distinct sums {0,1,2,4}. On the other hand, 12 is not in the sequence because {2} and {1,1} have the same sum.

Mathematica

msubs[s_]:=Join@@@Tuples[Table[Take[t, i], {t, Split[s]}, {i, 0, Length[t]}]];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], UnsameQ@@Total/@Select[msubs[primeMS[#]], SameQ@@#&]&]

Crossrefs

Knapsack partitions are counted by A108917, ranked by A299702.

The strong case is A353838, counted by A353837, complement A353839.

These partitions are counted by A353864.

The complete case is A353867, counted by A353865.

The complement is A354583.

A000041 counts partitions, strict A000009.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798 and A296150.

A073093 counts prime-power divisors.

A124010 gives prime signature, sorted A118914.

A300273 ranks collapsible partitions, counted by A275870.

A353832 represents the operation of taking run-sums of a partition.

A353836 counts partitions by number of distinct run-sums.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

A353863 counts partitions whose weak run-sums cover an initial interval.

Cf. A018818, A067340, A181819, A304442, A316413, A325862, A353833, A353835, A353861, A353931.

Sequence in context: A098240 A023805 A168186 * A085235 A364461 A160453

Adjacent sequences: A353863 A353864 A353865 * A353867 A353868 A353869

Keyword

nonn

Author

Gus Wiseman, Jun 06 2022

Status

approved