A346697 - OEIS

%I #8 Aug 02 2021 07:56:45

%S 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 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 3,6,19,8,2,5,20,4,21,1,5,9,4,7,22,5,4,1

%N Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of 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.

%F a(n) = A056239(n) - A346698(n).

%F a(n) = A316524(n) + A346698(n).

%F a(n odd omega) = A346699(n).

%F a(n even omega) = A346700(n).

%F A344616(n) = A346699(n) - A346700(n).

%e The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.

%e The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.

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

%t Table[Total[First/@Partition[Append[primeMS[n],0],2]],{n,100}]

%Y The version for standard compositions is A209281(n+1) (even: A346633).

%Y Subtracting the even version gives A316524 (reverse: A344616).

%Y The even version is A346698.

%Y The reverse version is A346699.

%Y The even reverse version is A346700.

%Y A000120 and A080791 count binary digits 1 and 0, with difference A145037.

%Y A000302 counts compositions with odd alternating sum, ranked by A053738.

%Y A001414 adds up prime factors, row sums of A027746.

%Y A029837 adds up parts of standard compositions (alternating: A124754).

%Y A056239 adds up prime indices, row sums of A112798.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A344606 counts alternating permutations of prime indices.

%Y Cf. A000070, A025047, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 01 2021