A344616 Alternating sum of the integer partition with Heinz number n.

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

Offset: 1

Author

Gus Wiseman, Jun 03 2021

Keywords

• nonn

Comments

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition.

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

Also the reverse-alternating sum of the prime indices of n.

Examples

			The partition (6,4,3,2,2) has Heinz number 4095 and conjugate (5,5,3,2,1,1), so a(4095) = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Positions of nonzeros are A000037.

Positions of 0's are A000290.

The version for prime factors is A071321 (reverse: A071322).

A version for compositions is A124754.

The version for prime multiplicities is A316523.

The reverse version is A316524, with sign A344617.

A000041 counts partitions of 2n with alternating sum 0.

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum.

A335433 ranks separable partitions.

A335448 ranks inseparable partitions.

A344606 counts wiggly permutations of prime indices with twins.

A344610 counts partitions by sum and positive reverse-alternating sum.

A344612 counts partitions by sum and reverse-alternating sum.

A344618 gives reverse-alternating sums of standard compositions.

Cf. A000070, A001222, A026424, A028260, A116406, A119899, A343938, A344607, A344608, A344609, A344619, A344653, A344739.

Programs

Maple

a:= n-> (l-> -add(l[i]*(-1)^i, i=1..nops(l)))(sort(map(
    i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..82);  # Alois P. Heinz, Jun 04 2021

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[ats[Reverse[primeMS[n]]],{n,100}]

Formula

a(n) = A257991(A122111(n)).

A057427(a(n)) = A049240(n).