A353866 - OEIS

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

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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 byA299702), 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 byA108917,ranked byA299702.

The strong case isA353838,counted byA353837,complementA353839.

These partitions are counted byA353864.

The complete case isA353867,counted byA353865.

The complement isA354583.

A000041counts partitions, strictA000009.

A001222counts prime factors, distinctA001221.

A056239adds up prime indices, row sums ofA112798andA296150.

A073093counts prime-power divisors.

A124010gives prime signature, sortedA118914.

A300273ranks collapsible partitions, counted byA275870.

A353832represents the operation of taking run-sums of a partition.

A353836counts partitions by number of distinct run-sums.

A353852ranks compositions with all distinct run-sums, counted byA353850.

A353863counts 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