cp's OEIS Frontend

A347460 Number of distinct possible alternating products of factorizations of n.

%I A347460 #9 Oct 27 2021 22:22:45
%S A347460 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,5,2,2,
%T A347460 2,7,1,2,2,6,1,5,1,4,4,2,1,8,2,4,2,4,1,5,2,6,2,2,1,10,1,2,4,6,2,5,1,4,
%U A347460 2,5,1,10,1,2,4,4,2,5,1,8,4,2,1,10,2,2
%N A347460 Number of distinct possible alternating products of factorizations of n.
%C A347460 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347460 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%e A347460 The a(n) alternating products for n = 1, 4, 8, 12, 24, 30, 36, 48, 60, 120:
%e A347460   1  4  8    12   24   30    36   48    60    120
%e A347460      1  2    3    6    10/3  9    12    15    30
%e A347460         1/2  3/4  8/3  5/6   4    16/3  20/3  40/3
%e A347460              1/3  2/3  3/10  1    3     15/4  15/2
%e A347460                   3/8  2/15  4/9  3/4   12/5  24/5
%e A347460                   1/6        1/4  1/3   3/5   10/3
%e A347460                              1/9  3/16  5/12  5/6
%e A347460                                   1/12  4/15  8/15
%e A347460                                         3/20  3/10
%e A347460                                         1/15  5/24
%e A347460                                               2/15
%e A347460                                               3/40
%e A347460                                               1/30
%t A347460 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347460 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347460 Table[Length[Union[altprod/@facs[n]]],{n,100}]
%Y A347460 Positions of 1's are 1 and A000040.
%Y A347460 Positions of 2's appear to be A001358.
%Y A347460 Positions of 3's appear to be A030078.
%Y A347460 Dominates A038548, the version for reverse-alternating product.
%Y A347460 Counting only integers gives A046951.
%Y A347460 The even-length case is A072670.
%Y A347460 The version for partitions (not factorizations) is A347461, reverse A347462.
%Y A347460 The odd-length case is A347708.
%Y A347460 The length-3 case is A347709.
%Y A347460 A001055 counts factorizations (strict A045778, ordered A074206).
%Y A347460 A056239 adds up prime indices, row sums of A112798.
%Y A347460 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347460 A108917 counts knapsack partitions, ranked by A299702.
%Y A347460 A276024 counts distinct positive subset-sums of partitions, strict A284640.
%Y A347460 A292886 counts knapsack factorizations, by sum A293627.
%Y A347460 A299701 counts distinct subset-sums of prime indices, positive A304793.
%Y A347460 A301957 counts distinct subset-products of prime indices.
%Y A347460 A304792 counts distinct subset-sums of partitions.
%Y A347460 Cf. A002033, A119620, A143823, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.
%K A347460 nonn
%O A347460 1,4
%A A347460 _Gus Wiseman_, Oct 06 2021