cp's OEIS Frontend

A345926 Number of distinct possible alternating sums of permutations of the multiset of prime indices of n.

%I A345926 #17 Aug 23 2021 13:39:00
%S A345926 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 A345926 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 A345926 2,3,1,3,1,2,2,2,2,3,1,2,1,2,1,4,2,2,2,2,1,3,2,2,2,2,2,2,1,2,2,3
%N A345926 Number of distinct possible alternating sums of permutations of the multiset of prime indices of n.
%C A345926 First differs from A096825 at a(90) = 3, A096825(90) = 4.
%C A345926 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A345926 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 indices is also the reverse-alternating sum of reversed prime indices.
%C A345926 Also the number of possible values of A056239(d) where d is a divisor of n with half as many prime factors (rounded up) as n.
%e A345926 Grouping the 12 permutations of {1,2,2,3} by alternating sum k gives:
%e A345926   k = -2: (1223) (1322) (2213) (2312)
%e A345926   k =  0: (1232) (2123) (2321) (3212)
%e A345926   k =  2: (2132) (2231) (3122) (3221)
%e A345926 so a(90) = 3.
%t A345926 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345926 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345926 Table[Length[Union[ats/@Permutations[primeMS[n]]]],{n,100}]
%o A345926 (Python)
%o A345926 from sympy import factorint, primepi
%o A345926 from sympy.utilities.iterables import multiset_combinations
%o A345926 def A345926(n):
%o A345926     fs = dict((primepi(a),b) for (a,b) in factorint(n).items())
%o A345926     return len(set(sum(d) for d in multiset_combinations(fs, (sum(fs.values())+1)//2))) # _Chai Wah Wu_, Aug 23 2021
%Y A345926 The version for prime factors instead of indices is A343943.
%Y A345926 A000005 counts divisors.
%Y A345926 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345926 A001414 adds up prime factors, row sums of A027746.
%Y A345926 A056239 adds up prime indices,  row sums of A112798.
%Y A345926 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345926 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345926 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345926 A345197 counts compositions by length and alternating sum.
%Y A345926 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345926 Cf. A008549, A032443, A083399, A096825, A239830, A344607, A344609, A344651, A345957, A345960, A345961.
%K A345926 nonn
%O A345926 1,6
%A A345926 _Gus Wiseman_, Jul 14 2021