A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 6, 4, 3, 2, 7, 3, 8, 4, 4, 5, 9, 3, 3, 6, 4, 5, 10, 4, 11, 3, 5, 7, 4, 3, 12, 8, 6, 4, 13, 5, 14, 6, 5, 9, 15, 4, 4, 4, 7, 7, 16, 4, 5, 5, 8, 10, 17, 4, 18, 11, 6, 3, 6, 6, 19, 8, 9, 5, 20, 4, 21, 12, 5, 9, 5, 7, 22, 5, 4

Offset: 1

Author

Gus Wiseman, Aug 03 2021

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 5 + 3 + 1 = 9.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 4 + 3 + 1 = 8.
		

Crossrefs

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

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

The even version is A346700.

The non-reverse version (multisets instead of partitions) is A346697.

The even non-reverse version is A346698.

A001414 adds up prime factors, row sums of A027746.

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, A124754, A329888, A341446, A344617, A345957, A345958, A346701, A346703.

Programs

Mathematica

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

Formula

a(n) = A056239(A346701(n)).

a(n) = A056239(n) - A346700(n).

a(n) = A344616(n) + A346700(n).

a(n odd omega) = A346697(n).

a(n even omega) = A346698(n).

A316524(n) = A346697(n) - A346698(n).