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approved

editing

proposed

A subsetsubsequence of A316413.

allocatedHeinz numbers of integer partitions whose mean A326567/A326568 foris Gusa Wisemanpart.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 105, 107, 109, 110, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181

1,1

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sequence of terms together with their prime indices begins:

2: {1}

3: {2}

4: {1,1}

5: {3}

7: {4}

8: {1,1,1}

9: {2,2}

11: {5}

13: {6}

16: {1,1,1,1}

17: {7}

19: {8}

23: {9}

25: {3,3}

27: {2,2,2}

29: {10}

30: {1,2,3}

31: {11}

32: {1,1,1,1,1}

37: {12}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], MemberQ[primeMS[#], Mean[primeMS[#]]]&]

A subset of A316413.

Complement of A327476.

The enumeration of these partitions by sum is given by A237984.

Subsets whose mean is a part are A065795.

Numbers whose binary indices include their mean are A327478.

Cf. A000016, A056239, A067538, A112798, A240850, A325706, A326567/A326568.

allocated

nonn

Gus Wiseman, Sep 13 2019

approved

editing

allocated for Gus Wiseman

allocated

approved