A122181 -id:A122181 - cp's OEIS frontend

A348610 Number of alternating ordered factorizations of n.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 12, 1, 3, 3, 6, 1, 11, 1, 7, 3, 3, 3, 15, 1, 3, 3, 12, 1, 11, 1, 6, 6, 3, 1, 23, 1, 6, 3, 6, 1, 12, 3, 12, 3, 3, 1, 28, 1, 3, 6, 12, 3, 11, 1, 6, 3, 11, 1, 33, 1, 3, 6, 6, 3, 11, 1, 23, 4, 3

Offset: 1

Gus Wiseman, Nov 05 2021

nonn

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

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The alternating ordered factorizations of n = 1, 6, 12, 16, 24, 30, 32, 36:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3                   3*6*2
                                                     6*2*3
                                                     2*3*2*3
                                                     3*2*3*2

Wikipedia, Alternating permutation

The additive version (compositions) is A025047 ranked by A345167.
The complementary additive version is A345192, ranked by A345168.
Dominated by A348611 (the anti-run version) at positions A122181.
The complement is counted by A348613.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A347463 counts ordered factorizations with integer alternating product.
A348379 counts factorizations w/ an alternating permutation.
A348380 counts factorizations w/o an alternating permutation.
Cf. A038548, A049774, A056986, A102726, A128761, A344614, A347050, A347464, A347706, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] == Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[ordfacs[n],wigQ]],{n,100}]

A349059 Number of weakly alternating ordered factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 18, 2, 3, 4, 8, 1, 11, 1, 16, 3, 3, 3, 22, 1, 3, 3, 18, 1, 11, 1, 8, 8, 3, 1, 38, 2, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 32, 1, 3, 8, 28, 3, 11, 1, 8, 3, 11, 1, 56, 1, 3, 8, 8, 3, 11, 1, 38, 8, 3

Offset: 1

Gus Wiseman, Dec 04 2021

nonn

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

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

			The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:
  (2)  (4)    (6)    (8)      (12)     (24)       (30)
       (2*2)  (2*3)  (2*4)    (2*6)    (3*8)      (5*6)
              (3*2)  (4*2)    (3*4)    (4*6)      (6*5)
                     (2*2*2)  (4*3)    (6*4)      (10*3)
                              (6*2)    (8*3)      (15*2)
                              (2*2*3)  (12*2)     (2*15)
                              (2*3*2)  (2*12)     (3*10)
                              (3*2*2)  (2*2*6)    (2*5*3)
                                       (2*4*3)    (3*2*5)
                                       (2*6*2)    (3*5*2)
                                       (3*2*4)    (5*2*3)
                                       (3*4*2)
                                       (4*2*3)
                                       (6*2*2)
                                       (2*2*2*3)
                                       (2*2*3*2)
                                       (2*3*2*2)
                                       (3*2*2*2)

The strong version for compositions is A025047, also A025048, A025049.
The strong case is A348610, complement A348613.
The version for compositions is A349052, complement A349053.
As compositions these are ranked by the complement of A349057.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating permutations of prime factors, w/ twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 = factorizations w/ alternating permutation, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
A349800 = weakly but not strongly alternating compositions, ranked A349799.
Cf. A003242, A122181, A138364, A339846, A339890, A345165, A345167, A345194, A347050, A347438, A347463, A347706.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]], {m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n], whkQ[#]||whkQ[-#]&]],{n,100}]

a(2^n) = A349052(n).

A122180 Number of ways to write n as n = xyz with 1 < x < y < z < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 1, 1

Offset: 1

Rick L. Shepherd, Aug 23 2006

nonn

x,y,z are distinct proper factors of n. See A122181 for n such that a(n) > 0.

If n has at most five divisors then a(n) = 0. - David A. Corneth, Oct 24 2024

			a(48) = 2 because 48 = 2*3*8 = 2*4*6, two products of three distinct proper factors of 48.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1001 terms from Antti Karttunen)
Index entries for sequences computed from exponents in factorization of n

Cf. A034836, A088432, A088433, A088434, A122179, A122181, A200214.

for(n=1,105, t=0; for(x=2,n-1, for(y=x+1,n-1, for(z=y+1,n-1, if(x*y*z==n, t++)))); print1(t,", "))

A122180(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xAntti Karttunen, Jul 08 2017

a(n) = {
	my(d = divisors(n));
	if(#d <= 5, return(0));
	my(res = 0, q);
	for(i = 2, #d,
		q = d[#d + 1 - i];
		if(d[i]^2 > q,
			return(res)
		);
		for(j = i + 1, #d,
			qj = q/d[j];
			if(qj <= d[j],
				next(2)
			);
			if(denominator(qj) == 1 && n % qj == 0,
				res++
			);
		);
	);
	res
} \\ David A. Corneth, Oct 24 2024

a(n) = A200214(n)/6. - Antti Karttunen, Jul 08 2017

A348611 Number of ordered factorizations of n with no adjacent equal factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 14, 1, 3, 3, 6, 1, 13, 1, 7, 3, 3, 3, 17, 1, 3, 3, 14, 1, 13, 1, 6, 6, 3, 1, 29, 1, 6, 3, 6, 1, 14, 3, 14, 3, 3, 1, 36, 1, 3, 6, 14, 3, 13, 1, 6, 3, 13, 1, 45, 1, 3, 6, 6, 3, 13, 1, 29, 4, 3

Offset: 1

Gus Wiseman, Nov 07 2021

nonn

First differs from A348610 at a(24) = 14, A348610(24) = 12.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.

			The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*3*4   2*3*5           2*3*6
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3   5*3*2           3*6*2
                             4*3*2                   6*2*3
                                                     6*3*2
                                                     2*3*2*3
                                                     3*2*3*2
Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - _Antti Karttunen_, Nov 12 2021

The additive version (compositions) is A003242, complement A261983.
The additive alternating version is A025047, ranked by A345167.
Factorizations without a permutation of this type are counted by A333487.
As compositions these are ranked by A333489, complement A348612.
Factorizations with a permutation of this type are counted by A335434.
The non-alternating additive version is A345195, ranked by A345169.
The alternating case is A348610, which is dominated at positions A122181.
The complement is counted by A348616.
A001055 counts factorizations, strict A045778, ordered A074206.
A325534 counts separable partitions, ranked by A335433.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
Cf. A001250, A138364, A336103, A347050, A347438, A347463, A347466, A347706, A348379, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],antirunQ]],{n,100}]

A348611(n, e=0) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d!=e), s += A348611(n/d, d))); (s)); \\ Antti Karttunen, Nov 12 2021

a(n) = A074206(n) - A348616(n).

A350139 Number of non-weakly alternating ordered factorizations of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 12, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 2, 0, 10, 0, 0, 0, 12, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

			The a(n) ordered factorizations for n = 24, 36, 48, 60:
  (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)
  (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)
           (2*3*3*2)  (6*4*2)    (5*4*3)
           (3*2*2*3)  (8*3*2)    (6*5*2)
                      (2*2*3*4)  (10*3*2)
                      (2*3*4*2)  (2*3*10)
                      (2*4*3*2)  (2*2*3*5)
                      (3*2*2*4)  (2*3*5*2)
                      (4*2*2*3)  (2*5*3*2)
                      (4*3*2*2)  (3*2*2*5)
                                 (5*2*2*3)
                                 (5*3*2*2)

Positions of nonzero terms are A122181.
The strong version for compositions is A345192, ranked by A345168.
The strong case is A348613, complement A348610.
The version for compositions is A349053, complement A349052.
As compositions with ones allowed these are ranked by A349057.
The complement is counted by A349059.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts weakly alternating compositions, ranked by A345167.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating perms of prime factors, with twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 counts factorizations w/ alternating perm, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n],!whkQ[#]&&!whkQ[-#]&]],{n,100}]

a(2^n) = A349053(n).

A348616 Number of ordered factorizations of n with adjacent equal factors.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 6, 1, 0, 1, 2, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 19, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 2, 18, 0, 0, 0, 2, 0, 0, 0, 31, 0, 0, 2, 2, 0, 0, 0, 19, 4, 0, 0, 8, 0, 0

Offset: 1

Gus Wiseman, Nov 08 2021

nonn

First differs from A348613 at a(24) = 6, A348613(24) = 8.

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

			The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
   2*2*3    2*2*6      6*6        15*2*2
   3*2*2    6*2*2      2*2*9      2*2*15
            2*2*2*3    3*3*4      2*2*3*5
            2*2*3*2    4*3*3      2*2*5*3
            2*3*2*2    9*2*2      3*2*2*5
            3*2*2*2    2*2*3*3    3*5*2*2
                       2*3*3*2    5*2*2*3
                       3*2*2*3    5*3*2*2
                       3*3*2*2
See also examples in A348611.

Positions of 0's are A005117.
Positions of 4's appear to be A030514.
Positions of 2's appear to be A054753.
Positions of 1's appear to be A168363.
The additive version (compositions) is A261983, complement A003242.
Factorizations with a permutation of this type are counted by A333487.
Factorizations without a permutation of this type are counted by A335434.
The complement is counted by A348611.
As compositions these are ranked by A348612, complement A333489.
Dominated by A348613 (non-alternating ordered factorizations).
A001055 counts factorizations, strict A045778, ordered A074206.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],!antirunQ[#]&]],{n,100}]

a(n) = A074206(n) - A348611(n).

A338112 Least number that is both the sum and product of n distinct positive integers.

Original entry on oeis.org

1, 3, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Rick L. Shepherd, Oct 10 2020

Each a(n) = n! except that a(2) = 1+2 = 3. For n > 0, only each integer >= A000217(n) is the sum of n distinct positive integers. For the integers that are products of these types, see below.

			a(1) = 1 because we define sums and products as sum(m) := prod(m) := m for all integers m in this case where these normally-binary operations only have one operand.
a(3) = 6 because 6 = 1+2+3 = 1*2*3 (with all the distinct positive integers the same in the sum and the product only for this term and a(1)).
a(5) = 120 because 120 = 1+2+3+4+110 (= ... = 22+23+24+25+26) = 1*2*3*4*5.

Cf. A000142, A000217.

Cf. Products of k distinct positive integers: A000027 (k=1), A020725 (k=2), A080257 (k=3), A122181 (k=4).

Array[If[# <= 2, (#^2 - #)/2 &[# + 1], #!] &, 22] (* Michael De Vlieger, Oct 15 2020 *)
With[{nn=30},Rest[CoefficientList[Series[x (2+x-x^2)/(2(1-x)),{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Aug 10 2021 *)

PARI
```
a(n) = if(n<1, , if(n==2, 3, n!))
```

a(n) = A000142(n) for n = 1 and n > 2; a(2) = 3.
a(n) = max(A000142(n), A000217(n)).
E.g.f.: x*(2 + x - x^2)/(2*(1 - x)). - Stefano Spezia, Oct 11 2020

cp's OEIS Frontend

A348610 Number of alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A349059 Number of weakly alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A122180 Number of ways to write n as n = x*y*z with 1 < x < y < z < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A348611 Number of ordered factorizations of n with no adjacent equal factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A350139 Number of non-weakly alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A348616 Number of ordered factorizations of n with adjacent equal factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A338112 Least number that is both the sum and product of n distinct positive integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A122180 Number of ways to write n as n = xyz with 1 < x < y < z < n.