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

Author

Gus Wiseman, Aug 03 2021

Keywords

• nonn

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.

Examples

			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.
		

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

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.

Programs

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}]

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

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

Extensions

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