A347049 -id:A347049

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A347463

Number of ordered factorizations of n with integer alternating product.

+10
26

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 7, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 11, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 20, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 26, 1, 1, 1, 4, 1, 1, 1, 35, 1, 1, 4, 4, 1, 1, 1, 20, 7, 1, 1, 8, 1, 1, 1, 6, 1, 8, 1, 4, 1, 1, 1, 32, 1, 4, 4, 18

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

An ordered factorization of n is a sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = A347048(n) + A347049(n).

EXAMPLE

The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:

4 8 12 16 24 32 36

2*2 4*2 6*2 4*4 12*2 8*4 6*6

2*2*2 2*2*3 8*2 2*2*6 16*2 12*3

3*2*2 2*2*4 3*2*4 2*2*8 18*2

2*4*2 4*2*3 2*4*4 2*2*9

4*2*2 6*2*2 4*2*4 2*3*6

2*2*2*2 4*4*2 2*6*3

8*2*2 3*2*6

2*2*4*2 3*3*4

4*2*2*2 3*6*2

2*2*2*2*2 4*3*3

6*2*3

6*3*2

9*2*2

2*2*3*3

2*3*3*2

3*2*2*3

3*3*2*2

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];

Table[Length[Select[Join@@Permutations/@facs[n], IntegerQ[altprod[#]]&]], {n, 100}]

PROG

(PARI) A347463(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if(d>1, A347463(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

CROSSREFS

Positions of 2's are A001248.

Positions of 1's are A005117.

The restriction to powers of 2 is A116406.

The even-length case is A347048

The odd-length case is A347049.

The unordered version is A347437, reciprocal A347439, reverse A347442.

The case of partitions is A347446, reverse A347445, ranked by A347457.

A001055 counts factorizations (strict A045778, ordered A074206).

A046099 counts factorizations with no alternating permutations.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A119620 counts partitions with alternating product 1, ranked by A028982.

A273013 counts ordered factorizations of n^2 with alternating product 1.

A339846 counts even-length factorizations, ordered A174725.

A339890 counts odd-length factorizations, ordered A174726.

A347438 counts factorizations with alternating product 1.

A347460 counts possible alternating products of factorizations.

Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 07 2021

EXTENSIONS

Data section extended up to a(100) by Antti Karttunen, Jul 28 2024

STATUS

approved

A347048

Number of even-length ordered factorizations of n with integer alternating product.

+10
3

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, 4, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 6, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 11, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 6, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 8, 0, 1, 1, 7, 0, 0, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,16

COMMENTS

An ordered factorization of n is a sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = A347463(n) - A347049(n).

EXAMPLE

The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:

4*4 8*4 6*6 12*4 8*8 24*4

8*2 16*2 12*3 24*2 16*4 48*2

2*2*2*2 2*2*4*2 18*2 2*2*6*2 32*2 3*2*8*2

4*2*2*2 2*2*3*3 3*2*4*2 2*2*4*4 4*2*6*2

2*3*3*2 4*2*3*2 2*2*8*2 6*2*4*2

3*2*2*3 6*2*2*2 2*4*4*2 8*2*3*2

3*3*2*2 4*2*2*4 12*2*2*2

4*2*4*2 2*2*12*2

4*4*2*2

8*2*2*2

2*2*2*2*2*2

MATHEMATICA

ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];

altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];

Table[Length[Select[ordfacs[n], EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 100}]

PROG

(PARI) A347048(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==numerator(ap), sumdiv(n, d, if(d>1, A347048(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

CROSSREFS

Positions of 0's are A005117 \ {2}.

The restriction to powers of 2 is A027306.

Heinz numbers of partitions of this type are A028260 /\ A347457.

Positions of 3's appear to be A030514.

Positions of 1's are 1 and A082293.

Allowing non-integer alternating product gives A174725, unordered A339846.

The odd-length version is A347049.

The unordered version is A347438, reverse A347439.

Allowing any length gives A347463.

Partitions of this type are counted by A347704, reverse A035363.

A001055 counts factorizations (strict A045778, ordered A074206).

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A119620 counts partitions with alternating product 1, ranked by A028982.

A273013 counts ordered factorizations of n^2 with alternating product 1.

A339890 counts odd-length factorizations, ordered A174726.

A347050 = factorizations with alternating permutation, complement A347706.

A347437 = factorizations with integer alternating product, reverse A347442.

A347446 = partitions with integer alternating product, reverse A347445.

A347460 counts possible alternating products of factorizations.

Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.