cp's OEIS Frontend

A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

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