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%I #13 Oct 22 2023 16:43:32
%S 1,1,1,2,1,1,1,3,2,1,1,3,1,1,1,5,1,3,1,3,1,1,1,3,2,1,3,3,1,1,1,7,1,1,
%T 1,8,1,1,1,3,1,1,1,3,3,1,1,8,2,3,1,3,1,4,1,3,1,1,1,3,1,1,3,11,1,1,1,3,
%U 1,1,1,11,1,1,3,3,1,1,1,8,5,1,1,3,1,1,1,3,1,4,1,3,1,1,1,9,1,3,3,8,1,1,1,3,1,1,1,12
%N Number of factorizations of n with integer reverse-alternating product.
%C A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
%H Antti Karttunen, <a href="/A347442/b347442.txt">Table of n, a(n) for n = 1..65537</a>
%F a(2^n) = A000041(n).
%e The a(n) factorizations for n = 4, 8, 16, 32, 36, 54, 64:
%e (4) (8) (16) (32) (36) (54) (64)
%e (2*2) (2*4) (2*8) (4*8) (6*6) (3*18) (8*8)
%e (2*2*2) (4*4) (2*16) (2*18) (2*3*9) (2*32)
%e (2*2*4) (2*2*8) (3*12) (3*3*6) (4*16)
%e (2*2*2*2) (2*4*4) (2*2*9) (2*4*8)
%e (2*2*2*4) (2*3*6) (4*4*4)
%e (2*2*2*2*2) (3*3*4) (2*2*16)
%e (2*2*3*3) (2*2*2*8)
%e (2*2*4*4)
%e (2*2*2*2*4)
%e (2*2*2*2*2*2)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
%t Table[Length[Select[facs[n],IntegerQ@*revaltprod]],{n,100}]
%o (PARI) A347442(n, m=n, ap=1, e=0) = if(1==n, 1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347442(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Oct 22 2023
%Y The restriction to powers of 2 is A000041, reverse A344607.
%Y Positions of 2's are A001248.
%Y Positions of 1's are A005117.
%Y Positions of non-1's are A013929.
%Y Allowing any alternating product <= 1 gives A339846.
%Y Allowing any alternating product > 1 gives A339890.
%Y The non-reverse version is A347437.
%Y The reciprocal version is A347438.
%Y The even-length case is A347439.
%Y Allowing any alternating product < 1 gives A347440.
%Y The odd-length case is A347441, ranked by A347453.
%Y The additive version is A347445, ranked by A347457.
%Y The non-reverse additive version is A347446, ranked by A347454.
%Y Allowing any alternating product >= 1 gives A347456.
%Y The ordered version is A347463.
%Y A038548 counts possible reverse-alternating products of factorizations.
%Y A071321 gives the alternating sum of prime factors (reverse: A071322).
%Y A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y Cf. A000290, A025047, A330972, A347443, A347449, A347451, A347458, A347459, A347460, A347462.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 08 2021
%E Data section extended up to a(108) by _Antti Karttunen_, Oct 22 2023
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