A324930 Total 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

Offset: 1

Author

Gus Wiseman, Mar 21 2019

Keywords

• nonn

Comments

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.

Examples

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

Crossrefs

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

Programs

Mathematica

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}]