login
Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).
3

%I #5 Mar 22 2019 00:32:24

%S 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,

%T 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,

%U 2,1,1,0,3,1,2,0,1,1,1,1,0,2,2,0,3,2,1

%N Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

%C 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.

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

%e 1: o

%e 2: (o)

%e 3: ((o))

%e 4: (oo)

%e 5: (((1)))

%e 6: (o(o))

%e 7: ((oo))

%e 8: (ooo)

%e 9: ((o)(o))

%e 10: (o((1)))

%e 11: (((2)))

%e 12: (oo(o))

%e 13: ((o(1)))

%e 14: (o(oo))

%e 15: ((o)((1)))

%e 16: (oooo)

%e 17: (((11)))

%e 18: (o(o)(o))

%e 19: ((ooo))

%e 20: (oo((1)))

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

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

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

%K nonn

%O 1,17

%A _Gus Wiseman_, Mar 21 2019