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A347439
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Number of factorizations of n with integer reciprocal alternating product.
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29
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1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 6, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 5
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OFFSET
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1,16
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COMMENTS
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All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
The value of a(n) does not depend solely on the prime signature of n. See the example comparing a(144) and a(400). - Antti Karttunen, Jul 28 2024
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LINKS
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FORMULA
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EXAMPLE
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The a(n) factorizations for
n = 16, 36, 64, 72, 128, 144:
a(n) = 3, 4, 6, 5, 7, 11
--------------------------------------------------------------------------------
2*8 6*6 8*8 2*36 2*64 2*72
4*4 2*18 2*32 3*24 4*32 3*48
2*2*2*2 3*12 4*16 6*12 8*16 4*36
2*2*3*3 2*2*2*8 2*2*3*6 2*2*4*8 6*24
2*2*4*4 2*3*3*4 2*4*4*4 12*12
2*2*2*2*2*2 2*2*2*16 2*2*6*6
2*2*2*2*2*4 2*3*3*8
3*3*4*4
2*2*2*18
2*2*3*12
2*2*2*2*3*3
For n=400, there are 12 such factorizations:
2*200
4*100
5*80
10*40
20*20
2*2*2*50
2*2*5*20
2*2*10*10
2*4*5*10
2*5*5*8
4*4*5*5
2*2*2*2*5*5.
Note that 400 = 2^4 * 5^2 has the same prime signature as 144 = 2^4 * 3^2. 400 = 2*4*5*10 is the factorization for which there is no analogous factorization of 144, as 2*3*4*6 doesn't satisfy the condition of having an integer reciprocal alternating product.
(End)
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i, {i, Length[q]}];
Table[Length[Select[facs[n], IntegerQ[recaltprod[#]]&]], {n, 100}]
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PROG
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(PARI) A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
(PARI) A347439(n, m=0, ap=1, e=1) = if(1==n, 1==denominator(ap), sumdiv(n, d, if(d>1 && d>=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
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CROSSREFS
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Positions of 0's are A005117 \ {1}.
Positions of non-0's are 1 and A013929.
Positions of 1's are 1 and A082293.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reciprocal version is A347437.
Allowing any alternating product < 1 gives A347440.
The non-reciprocal reverse version is A347442.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347459, non-reciprocal A347458.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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