proposed

approved

editing

proposed

The sequence of all multiset multisystem multisystems finite multisets of finite multisets of finite multisets of positive integers begins (o is the empty multiset) begins:

21: ((o)(oo))

22: (o((2)))

23: (((1)(1)))

24: (ooo(o))

25: (((1))((1)))

26: (o(o(1)))

27: ((o)(o)(o))

28: (oo(oo))

29: ((o(2)))

30: (o(o)((1)))

Cf. A000081, A000720, A001222, A003963, A050338, A056239, A112798, A301595, A302242, A318564, A318565, A318566, A324928.

allocated for Gus WisemanTotal weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 2, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 0, 2, 1, 3, 2, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 3, 2, 1

1,17

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 finite multiset of finite multisets of finite multisets of positive integers with MMM number n is obtained by factoring n into prime numbers, then factoring each of their prime indices into prime numbers, then factoring each of their prime indices into prime numbers, and finally taking their prime indices.

The sequence of all multiset multisystem multisystems (o is the empty multiset) begins:

1: o

2: (o)

3: ((o))

4: (oo)

5: (((1)))

6: (o(o))

7: ((oo))

8: (ooo)

9: ((o)(o))

10: (o((1)))

11: (((2)))

12: (oo(o))

13: ((o(1)))

14: (o(oo))

15: ((o)((1)))

16: (oooo)

17: (((11)))

18: (o(o)(o))

19: ((ooo))

20: (oo((1)))

21: ((o)(oo))

22: (o((2)))

23: (((1)(1)))

24: (ooo(o))

25: (((1))((1)))

26: (o(o(1)))

27: ((o)(o)(o))

28: (oo(oo))

29: ((o(2)))

30: (o(o)((1)))

fi[n_]:=If[n==1, {}, FactorInteger[n]];

Table[Total[Cases[fi[n], {p_, k_}:>k*Total[Cases[fi[PrimePi[p]], {q_, j_}:>j*PrimeOmega[PrimePi[q]]]]]], {n, 60}]

Cf. A000081, A000720, A001222, A003963, A050338, A056239, A112798, A301595, A302242, A318564, A318565, A318566, A324928.

allocated

nonn

Gus Wiseman, Mar 21 2019

approved

editing

allocated for Gus Wiseman

allocated

approved