A298540 -id:A298540

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A326535

MM-numbers of multiset partitions where each part has a different sum.

+10
16

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

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OFFSET

1,2

COMMENTS

First differs from A298540 in having 187.

These are numbers where each prime index has a different sum of prime indices. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

LINKS

Table of n, a(n) for n=1..65.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

EXAMPLE

The sequence of multiset partitions where each part has a different sum, preceded by their MM-numbers, begins:

1: {}

2: {{}}

3: {{1}}

5: {{2}}

6: {{},{1}}

7: {{1,1}}

10: {{},{2}}

11: {{3}}

13: {{1,2}}

14: {{},{1,1}}

15: {{1},{2}}

17: {{4}}

19: {{1,1,1}}

21: {{1},{1,1}}

22: {{},{3}}

23: {{2,2}}

26: {{},{1,2}}

29: {{1,3}}

30: {{},{1},{2}}

31: {{5}}

MATHEMATICA

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

Select[Range[100], UnsameQ@@Total/@primeMS/@primeMS[#]&]

CROSSREFS

Cf. A038041, A112798, A275780, A302242, A320324, A321469, A326519, A326533, A326534, A326536, A326537.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 12 2019

STATUS

approved

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