A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.

0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 4, 3, 2, 0, 2, 0, 1, 4, 5, 0, 3, 3, 6, 2, 1, 0, 2, 0, 2, 5, 7, 4, 3, 0, 8, 6, 4, 0, 2, 0, 1, 2, 9, 0, 2, 4, 3, 7, 1, 0, 4, 5, 5, 8, 10, 0, 4, 0, 11, 2, 3, 6, 2, 0, 1, 9, 3, 0, 3, 0, 12, 3, 1, 5, 2, 0, 2, 4, 13, 0, 5, 7, 14, 10, 6, 0, 5, 6, 1, 11, 15, 8, 4, 0, 4, 2, 4, 0, 2, 0, 7, 3

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 = 4.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 3 + 4 = 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Subtracting from the odd version gives A316524 (reverse: A344616).

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

The odd version is A346697.

The even reverse version is A346699.

The reverse version is A346700.

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

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, A000097, A025047, A035363, A120452, A344617, A344653, A344654, A345957, A345958.

Programs

Mathematica

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

PARI

A346698(n) = if(1==n,0,my(f=factor(n),s=0,p=0); for(k=1,#f~,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ Antti Karttunen, Nov 30 2021

Formula

a(n) = A056239(n) - A346697(n).

a(n) = A346697(n) - A316524(n).

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

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

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

Extensions

Data section extended up to 105 terms by Antti Karttunen, Nov 30 2021