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%I #15 Aug 23 2021 13:37:26
%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,3,1,1,2,2,
%T 2,3,1,2,2,2,1,3,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,4,1,2,2,1,2,3,1,2,
%U 2,3,1,3,1,2,2,2,2,3,1,2,1,2,1,4,2,2,2
%N Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.
%C First differs from A096825 at a(525) = 3, A096825(525) = 4.
%C First differs from A345926 at a(90) = 4, A345926(90) = 3.
%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime factors is also the reverse-alternating sum of reversed prime factors.
%C Also the number of distinct "sums of prime factors" of divisors d|n such that bigomega(d) = bigomega(n)/2 rounded up.
%e The divisors of 525 with 2 prime factors are: 15, 21, 25, 35, with prime factors {3,5}, {3,7}, {5,5}, {5,7}, with distinct sums {8,10,12}, so a(525) = 3.
%t prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t Table[Length[Union[Total/@Subsets[prifac[n],{Ceiling[PrimeOmega[n]/2]}]]],{n,100}]
%o (Python)
%o from sympy import factorint
%o from sympy.utilities.iterables import multiset_combinations
%o def A343943(n):
%o fs = factorint(n)
%o return len(set(sum(d) for d in multiset_combinations(fs,(sum(fs.values())+1)//2))) # _Chai Wah Wu_, Aug 23 2021
%Y The half-length submultisets are counted by A114921.
%Y Including all multisets of prime factors gives A305611(n) + 1.
%Y The strict rounded version appears to be counted by A342343.
%Y The version for prime indices instead of prime factors is A345926.
%Y A000005 counts divisors, which add up to A000203.
%Y A001414 adds up prime factors, row sums of A027746.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A071321 gives the alternating sum of prime factors (reverse: A071322).
%Y A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A108917 counts knapsack partitions, ranked by A299702.
%Y A276024 and A299701 count positive subset-sums of partitions.
%Y A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A334968 counts subsequence-sums of standard compositions.
%Y Cf. A008549, A032443, A083399, A096825, A344609, A345957.
%K nonn
%O 1,6
%A _Gus Wiseman_, Aug 19 2021
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