A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 3, 1, 2, 1, 0, 2, 0, 2, 2, 1, 3, 3, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 4, 3, 2, 1, 0, 3, 3, 2, 2, 1, 0, 3, 0, 1, 2, 3, 3, 2, 0, 1, 2, 3, 0, 3, 0, 1, 3, 1, 4, 2, 0, 2, 4, 1, 0, 3, 3, 1, 2, 2, 0, 3, 4, 1, 2, 1, 3, 3, 0, 4, 2, 4, 0, 2, 0, 2, 3

Offset

1,9

Comments

First differs from A334107 at a(64) = 3, A334107(64) = 2.

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

Antti Karttunen, Table of n, a(n) for n = 1..10201

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences related to Heinz numbers

Formula

a(n) = A056239(n) - A346699(n).

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

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

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

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

a(n) = A056239(A329888(n)). - Gus Wiseman and Antti Karttunen, Oct 13 2021

Example

The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 3 + 1 = 4.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 3 + 2 + 1 = 6.

Mathematica

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

Prog

(PARI) A346700(n) = if(1==n, 0, my(f=factor(n), s=0, p=0); forstep(k=#f~, 1, -1, while(f[k, 2], s += (p%2)*primepi(f[k, 1]); f[k, 2]--; p++)); (s)); \\ Antti Karttunen, Sep 21 2021

Crossrefs

Sum of prime indices of A329888(n).

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

The unreversed version for standard compositions is A346633.

The odd non-reverse version is A346697.

The non-reverse version (multisets instead of partitions) is A346698.

The odd version is A346699.

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, A124754, A209281, A329888, A341446, A344617, A344653, A345957, A345958, A346701, A346702, A346704.

Sequence in context: A062756 A360676 A334107 * A301574 A272728 A174695

Adjacent sequences: A346697 A346698 A346699 * A346701 A346702 A346703

Keyword

nonn

Author

Gus Wiseman, Aug 03 2021

Extensions

Data section extended up to 105 terms by Antti Karttunen, Sep 21 2021

Status

approved