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

1,3

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.

Links

Table of n, a(n) for n=1..81.

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).

Example

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.

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}]

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.

Sequence in context: A353420 A123021 A317056 * A360679 A339894 A355140

Adjacent sequences: A346696 A346697 A346698 * A346700 A346701 A346702

Keyword

nonn

Author

Gus Wiseman, Aug 03 2021

Status

approved