%I #12 Jun 04 2021 16:11:12

%S 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,

%T 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,

%U 0,3,4,19,7,7,2,20,1,21,11,2,8,1,5,22,3,0,12

%N Alternating sum of the integer partition with Heinz number n.

%C 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.

%C 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.

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

%H Alois P. Heinz, <a href="/A344616/b344616.txt">Table of n, a(n) for n = 1..20000</a>

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

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

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

%p a:= n-> (l-> -add(l[i]*(-1)^i, i=1..nops(l)))(sort(map(

%p i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):

%p seq(a(n), n=1..82); # _Alois P. Heinz_, Jun 04 2021

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Table[ats[Reverse[primeMS[n]]],{n,100}]

%Y Positions of nonzeros are A000037.

%Y Positions of 0's are A000290.

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

%Y A version for compositions is A124754.

%Y The version for prime multiplicities is A316523.

%Y The reverse version is A316524, with sign A344617.

%Y A000041 counts partitions of 2n with alternating sum 0.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A103919 counts partitions by sum and alternating sum.

%Y A335433 ranks separable partitions.

%Y A335448 ranks inseparable partitions.

%Y A344606 counts wiggly permutations of prime indices with twins.

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

%Y A344612 counts partitions by sum and reverse-alternating sum.

%Y A344618 gives reverse-alternating sums of standard compositions.

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Jun 03 2021