cp's OEIS Frontend

A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

%I A346697 #8 Aug 02 2021 07:56:45
%S A346697 0,1,2,1,3,1,4,2,2,1,5,3,6,1,2,2,7,3,8,4,2,1,9,2,3,1,4,5,10,4,11,3,2,
%T A346697 1,3,3,12,1,2,2,13,5,14,6,5,1,15,4,4,4,2,7,16,3,3,2,2,1,17,3,18,1,6,3,
%U A346697 3,6,19,8,2,5,20,4,21,1,5,9,4,7,22,5,4,1
%N A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.
%C A346697 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.
%F A346697 a(n) = A056239(n) - A346698(n).
%F A346697 a(n) = A316524(n) + A346698(n).
%F A346697 a(n odd omega) = A346699(n).
%F A346697 a(n even omega) = A346700(n).
%F A346697 A344616(n) = A346699(n) - A346700(n).
%e A346697 The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
%e A346697 The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.
%t A346697 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A346697 Table[Total[First/@Partition[Append[primeMS[n],0],2]],{n,100}]
%Y A346697 The version for standard compositions is A209281(n+1) (even: A346633).
%Y A346697 Subtracting the even version gives A316524 (reverse: A344616).
%Y A346697 The even version is A346698.
%Y A346697 The reverse version is A346699.
%Y A346697 The even reverse version is A346700.
%Y A346697 A000120 and A080791 count binary digits 1 and 0, with difference A145037.
%Y A346697 A000302 counts compositions with odd alternating sum, ranked by A053738.
%Y A346697 A001414 adds up prime factors, row sums of A027746.
%Y A346697 A029837 adds up parts of standard compositions (alternating: A124754).
%Y A346697 A056239 adds up prime indices, row sums of A112798.
%Y A346697 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A346697 A325534 counts separable partitions, ranked by A335433.
%Y A346697 A325535 counts inseparable partitions, ranked by A335448.
%Y A346697 A344606 counts alternating permutations of prime indices.
%Y A346697 Cf. A000070, A025047, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.
%K A346697 nonn
%O A346697 1,3
%A A346697 _Gus Wiseman_, Aug 01 2021