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

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

Offset: 1

Author

Gus Wiseman, Aug 01 2021

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.

Examples

			The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.
		

Crossrefs

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

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

The even version is A346698.

The reverse version is A346699.

The even reverse version is A346700.

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

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

A001414 adds up prime factors, row sums of A027746.

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

A056239 adds up prime indices, row sums of A112798.

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

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344606 counts alternating permutations of prime indices.

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Partition[Append[primeMS[n],0],2]],{n,100}]

Formula

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

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

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

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

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