A326537 - OEIS

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A326537

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

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110

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OFFSET

1,2

COMMENTS

These are numbers where each prime index has a different average 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 average, 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}}

22: {{},{3}}

23: {{2,2}}

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

29: {{1,3}}

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

31: {{5}}

33: {{1},{3}}

MATHEMATICA

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

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

CROSSREFS

Cf. A038041, A051293, A112798, A302242, A320324, A326513, A326516, A326521, A326533, A326534, A326535, A326536.

Sequence in context: A325399 A167171 A087008 * A302798 A294472 A077337

Adjacent sequences: A326534 A326535 A326536 * A326538 A326539 A326540

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 12 2019

STATUS

approved