A119620 -id:A119620 - cp's OEIS frontend

A100824 Number of partitions of n with at most one odd part.

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742

Offset: 0

Vladeta Jovovic, Jan 13 2005

From Gus Wiseman, Jan 21 2022: (Start)
Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
  1  11  21   22    32     33      43       44        54
         111  1111  221    2211    331      2222      441
                    2111   111111  2221     3311      3222
                    11111          3211     221111    3321
                                   22111    11111111  4311
                                   211111             22221
                                   1111111            33111
                                                      222111
                                                      321111
                                                      2211111
                                                      21111111
                                                      111111111
(End)

			From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)     (9)
            (21)  (22)  (32)   (42)   (43)    (44)    (54)
                        (41)   (222)  (52)    (62)    (63)
                        (221)         (61)    (422)   (72)
                                      (322)   (2222)  (81)
                                      (421)           (432)
                                      (2221)          (441)
                                                      (522)
                                                      (621)
                                                      (3222)
                                                      (4221)
                                                      (22221)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A008951, A000070, A000097, A000098, A000710.
The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
These partitions are ranked by A349150, conjugate A349151.
A000041 = integer partitions, strict A000009.
A027187 = partitions of even length, strict A067661, ranked by A028260.
A027193 = partitions of odd length, ranked by A026424.
A058695 = partitions of odd numbers.
A103919 = partitions by sum and alternating sum (reverse: A344612).
A277103 = partitions with the same number of odd parts as their conjugate.
Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.

seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)

a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

A347465 Numbers whose multiset of prime indices has alternating product > 1.

Original entry on oeis.org

3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127

Offset: 1

Gus Wiseman, Sep 27 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
All terms have odd bigomega (A001222).
Also Heinz numbers integer partitions with reverse-alternating product > 1.

			The terms and their prime indices begin:
      3: {2}         37: {12}            68: {1,1,7}
      5: {3}         41: {13}            70: {1,3,4}
      7: {4}         42: {1,2,4}         71: {20}
     11: {5}         43: {14}            73: {21}
     12: {1,1,2}     44: {1,1,5}         75: {2,3,3}
     13: {6}         45: {2,2,3}         76: {1,1,8}
     17: {7}         47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     52: {1,1,6}         80: {1,1,1,1,3}
     23: {9}         53: {16}            83: {23}
     27: {2,2,2}     59: {17}            89: {24}
     28: {1,1,4}     61: {18}            92: {1,1,9}
     29: {10}        63: {2,2,4}         97: {25}
     30: {1,2,3}     66: {1,2,5}         99: {2,2,5}
     31: {11}        67: {19}           101: {26}

The squarefree case is A030059 without 2.
The reverse version is A028983, counted by A119620.
The opposite version (< 1 instead of > 1) is A119899.
Factorizations of this type are counted by A339890, reverse A347705.
The weak version (>= 1 instead of > 1) is A344609.
Partitions of this type are counted by A347449, reverse A347448.
The complement is A347450, counted by A339846 or A347443.
Allowing any integer reverse-alternating product gives A347454.
Allowing any integer alternating product gives A347457.
A335433 ranks inseparable partitions, complement A335448.
A347446 counts partitions with integer alternating product, reverse A347445.
Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],altprod[primeMS[#]]>1&]

A347705 Number of factorizations of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 8, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 12 2021

nonn

A factorization of n is a weakly increasing 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)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:
  2   6     8       12      24        30      48          60
      2*3   2*4     2*6     3*8       5*6     6*8         2*30
            2*2*2   3*4     4*6       2*15    2*24        3*20
                    2*2*3   2*12      3*10    3*16        4*15
                            2*2*6     2*3*5   4*12        5*12
                            2*3*4             2*3*8       6*10
                            2*2*2*3           2*4*6       2*5*6
                                              3*4*4       3*4*5
                                              2*2*12      2*2*15
                                              2*2*2*6     2*3*10
                                              2*2*3*4     2*2*3*5
                                              2*2*2*2*3

Positions of 1's are A000430.
The weak version (>= instead of >) is A001055, non-reverse A347456.
The non-reverse version is A339890, strict A347447.
The version for reverse-alternating product 1 is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The even-length case is A347440, also the opposite reverse version.
Allowing any integer rev-alt product gives A347442, non-reverse A347437.
The version for partitions is A347449, non-reverse A347448.
A001055 counts factorizations (strict A045778, ordered A074206).
A038548 counts possible rev-alt products of factorizations, integer A046951.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A292886 counts knapsack factorizations, by sum A293627.
A347707 counts possible integer reverse-alternating products of partitions.
Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[q[[-i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],revaltprod[#]>1&]],{n,100}]

a(n) = A001055(n) - A347438(n).

A347464 Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 6, 2, 5, 1, 26, 1, 5, 5, 20, 1, 26, 1, 26, 5, 5, 1, 134, 2, 5, 6, 26, 1, 73, 1, 70, 5, 5, 5, 230, 1, 5, 5, 134, 1, 73, 1, 26, 26, 5, 1, 670, 2, 26, 5, 26, 1, 134, 5, 134, 5, 5, 1, 686, 1, 5, 26, 252, 5, 73, 1, 26, 5, 73, 1, 1714, 1, 5, 26

Offset: 1

Gus Wiseman, Sep 23 2021

nonn

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)).
Also the number of ordered pairs of ordered factorizations of n, both of the same length.
Note that the version for all n (not just squares) is 0 except at perfect squares.

			The a(12) = 26 ordered factorizations:
  (2*2*6*6)      (3*2*4*6)      (6*2*2*6)  (4*2*3*6)  (12*12)
  (2*3*6*4)      (3*3*4*4)      (6*3*2*4)  (4*3*3*4)
  (2*4*6*3)      (3*4*4*3)      (6*4*2*3)  (4*4*3*3)
  (2*6*6*2)      (3*6*4*2)      (6*6*2*2)  (4*6*3*2)
  (2*2*2*2*3*3)  (3*2*2*2*2*3)
  (2*2*2*3*3*2)  (3*2*2*3*2*2)
  (2*2*3*2*2*3)  (3*3*2*2*2*2)
  (2*2*3*3*2*2)
  (2*3*2*2*3*2)
  (2*3*3*2*2*2)
For example, the ordered factorization 6*3*2*4 = 144 has alternating product 6/3*2/4 = 1, so is counted under a(12).

Index entries for sequences computed from exponents in factorization of n

Positions of 1's are A008578 (1 and A000040).
The restriction to powers of 2 is A000984.
Positions of 2's are A001248.
The not necessarily even-length version is A273013.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A027187 counts even-length partitions.
A074206 counts ordered factorizations.
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A347706.
A347438 counts factorizations with alternating product 1.
A347457 ranks partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347466 counts factorizations of n^2.
Cf. A062312, A339890, A347437, A347439, A347440, A347442, A347456, A347459, A347463, A347705.

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^2],EvenQ[Length[#]]&&altprod[#]==1&]],{n,100}]

A347464aux(n, k=0, t=1) = if(1==n, (0==k)&&(1==t), my(s=0); fordiv(n, d, if((d>1), s += A347464aux(n/d, 1-k, t*(d^((-1)^k))))); (s));
A347464(n) = A347464aux(n^2); \\ Antti Karttunen, Oct 30 2021

A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347050 at a(216) = 28, A347050(216) = 27.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.

			The a(216) = 28 factorizations:
  (2*2*2*3*3*3)  (2*2*2*3*9)  (2*2*6*9)   (3*8*9)   (3*72)   (216)
                 (2*2*3*3*6)  (2*3*4*9)   (4*6*9)   (4*54)
                 (2*3*3*3*4)  (2*3*6*6)   (2*2*54)  (6*36)
                              (3*3*4*6)   (2*3*36)  (8*27)
                              (2*2*3*18)  (2*4*27)  (9*24)
                              (2*3*3*12)  (2*6*18)  (12*18)
                                          (2*9*12)  (2*108)
                                          (3*3*24)
                                          (3*4*18)
                                          (3*6*12)
The a(270) = 20 factorizations:
  (2*3*3*3*5)  (2*3*5*9)   (5*6*9)   (3*90)   (270)
               (3*3*5*6)   (2*3*45)  (5*54)
               (2*3*3*15)  (2*5*27)  (6*45)
                           (2*9*15)  (9*30)
                           (3*3*30)  (10*27)
                           (3*5*18)  (15*18)
                           (3*6*15)  (2*135)
                           (3*9*10)

Positions of 1's are 1 and A000040.
Not requiring separability gives A010052 for n > 1.
Positions of 2's are A323644.
Partitions of this type are counted by A325534(n) + A000035(n + 1).
Partitions of this type are ranked by A335433 \/ A001248.
Partitions not of this type are counted by A325535(n) - A000035(n + 1).
Partitions not of this type are ranked by A345193 = A335448 \ A001248.
Not allowing twins gives A335434, complement A333487,
The case with an alternating permutation is A347050, no twins A348379.
The case without an alternating permutation is A347706, no twins A348380.
The complement is counted by A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A003242 counts anti-run compositions, ranked by A333489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A119620, A138364, A336103, A344654, A344740, A347437, A347438, A347456, A347458, A348382.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
Table[Length[Select[facs[n],MatchQ[#,{x_,x_}]||sepQ[#]&]],{n,100}]

a(n > 1) = A335434(n) + A010052(n), where A010052(n) = 1 if n is a perfect square, otherwise 0.

A363623 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-weighted alternating sum k (leading and trailing 0's omitted).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 0, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 0, 3, 0, 1, 1, 1, 1, 3, 2, 0, 3, 1, 2, 0, 1, 0, 1, 2, 5, 1, 0, 3, 1, 2, 2, 2, 1, 1, 0, 1, 0, 1, 2, 5, 3, 0, 4, 2, 2, 0, 3, 2, 1, 3, 0, 0, 1, 0, 1, 1, 1, 1, 7, 2, 0, 4, 1, 5, 2, 3, 1, 3, 0, 2, 3, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 1

Offset: 0

Gus Wiseman, Jun 15 2023

We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}. For example:
- (3,3,2,1,1) has reverse-weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8.
- (1,2,2,3) has reverse-weighted alternating sum -1*3 + 2*2 - 3*2 + 4*1 = -1.

			Triangle begins:
  1
  1
  1  1
  1  2
  2  0  1  2
  2  1  1  1  1  1
  3  1  0  3  0  1  1  1  1
  3  2  0  3  1  2  0  1  0  1  2
  5  1  0  3  1  2  2  2  1  1  0  1  0  1  2
  5  3  0  4  2  2  0  3  2  1  3  0  0  1  0  1  1  1  1
Row n = 6 counts the following partitions:
  k=3       k=4       k=6       k=8      k=9   k=10    k=11
--------------------------------------------------------------
  (33)      (222)  .  (6)    .  (21111)  (51)  (3111)  (411)
  (2211)              (42)
  (111111)            (321)

Row sums are A000041.
Column k = floor((n+1)/2) is A119620.
The unweighted version is A344612 aerated, reverse A103919.
The corresponding rank statistic is A363620, reverse A363619.
The reverse version is A363622.
A053632 counts compositions by weighted sum.
A264034 counts partitions by weighted sum, reverse A358194.
A316524 gives alternating sum of prime indices, reverse A344616.
A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
Cf. A008284, A067538, A222855, A222970, A318283, A320387, A360672, A360675, A362559, A363532, A363621, A363626.

revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
Table[Length[Select[IntegerPartitions[n],revaltwtsum[#]==k&]],{n,0,15},{k,Floor[(n+1)/2],Ceiling[n*(n+1)/4]}]

A347447 Number of strict factorizations of n with alternating product > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 23 2021

nonn

A strict factorization of n is an increasing sequence of distinct 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)).
All such factorizations must have odd length.

			The a(720) = 30 factorizations:
  (2*4*90)     (3*4*60)   (4*5*36)   (5*6*24)  (6*8*15)   (8*9*10)  (720)
  (2*5*72)     (3*5*48)   (4*6*30)   (5*8*18)  (6*10*12)
  (2*6*60)     (3*6*40)   (4*9*20)   (5*9*16)
  (2*8*45)     (3*8*30)   (4*10*18)
  (2*9*40)     (3*10*24)  (4*12*15)
  (2*10*36)    (3*12*20)
  (2*12*30)    (3*15*16)
  (2*15*24)
  (2*18*20)
  (2*3*120)
  (2*3*4*5*6)

Allowing any alternating product gives A045778.
The reverse additive version (or restriction to powers of 2) is A067659.
The non-strict version is A339890.
Allowing equal parts and any alternating product < 1 gives A347440.
Allowing equal parts and any alternating product >= 1 gives A347456.
A046099 counts factorizations with no alternating permutations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339846 counts even-length factorizations.
A347437 counts factorizations with integer alternating product.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000009, A005117, A030059, A119620, A119899, A330972, A344608, A347438, A347439, A347442, A347463.

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[facs[n],UnsameQ@@#&&altprod[#]>1&]],{n,100}]

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 1, 2, 1, 4, 2, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 3, 1, 2, 1, 3, 2

Offset: 1

Gus Wiseman, Oct 11 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180

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

The version for partitions is A028310, reverse A347707.
Positions of 1's appear to be A037143 \ {1}.
The even-length version for n > 1 is A072670, strict A211159.
Counting only integers appears to give A293234, with evens A046951.
This is the odd-length case of A347460, reverse A038548.
The any-length version for partitions is A347461, reverse A347462.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A276024 counts distinct positive subset-sums of partitions.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347050 = factorizations w/ an alternating permutation, complement A347706.
A347441 counts odd-length factorizations with integer alternating product.
Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

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[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A347704 Number of even-length integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 6, 4, 11, 8, 18, 13, 33, 22, 49, 38, 79, 58, 122, 90, 186, 139, 268, 206, 402, 304, 569, 448, 817, 636, 1152, 907, 1612, 1283, 2220, 1791, 3071, 2468, 4162, 3409, 5655, 4634, 7597, 6283, 10171, 8478, 13491, 11336, 17906, 15088, 23513, 20012

Offset: 0

Gus Wiseman, Sep 17 2021

nonn

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

			The a(2) = 1 through a(9) = 8 partitions:
  (11)  (21)  (22)    (41)    (33)      (61)      (44)        (63)
              (31)    (2111)  (42)      (2221)    (62)        (81)
              (1111)          (51)      (4111)    (71)        (3321)
                              (2211)    (211111)  (2222)      (4221)
                              (3111)              (3221)      (6111)
                              (111111)            (3311)      (222111)
                                                  (4211)      (411111)
                                                  (5111)      (21111111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)

Allowing any alternating product >= 1 gives A000041, reverse A344607.
Allowing any alternating product gives A027187, odd bisection A236914.
The Heinz numbers of these partitions are given by A028260 /\ A347457.
The reverse and reciprocal versions are both A035363.
The multiplicative version (factorizations) is A347438, reverse A347439.
The odd-length instead of even-length version is A347444.
Allowing any length gives A347446.
A034008 counts even-length compositions, ranked by A053754.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

A347452 Heinz numbers of integer partitions whose sum is 3/2 their length, rounded down.

Original entry on oeis.org

1, 2, 6, 12, 36, 40, 72, 80, 216, 224, 240, 432, 448, 480, 1296, 1344, 1408, 1440, 1600, 2592, 2688, 2816, 2880, 3200, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 13312, 15552, 16128, 16896, 17280, 17920, 19200, 34816, 39936, 46656, 48384, 50176, 50688, 51840

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

Also numbers whose sum of prime indices is 3/2 their number, rounded down, where a prime index of n is a number m such that prime(m) divides n.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The sequence contains n iff A056239(n) = floor(3*A001222(n)/2). Here, A056239 adds up prime indices, and A001222 counts them with multiplicity.
Counting the partitions with these Heinz numbers gives A119620 with zeros interspersed every three terms.

			The initial terms and their prime indices:
      1: {}
      2: {1}
      6: {1,2}
     12: {1,1,2}
     36: {1,1,2,2}
     40: {1,1,1,3}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    240: {1,1,1,1,2,3}
    432: {1,1,1,1,2,2,2}
    448: {1,1,1,1,1,1,4}
    480: {1,1,1,1,1,2,3}
   1296: {1,1,1,1,2,2,2,2}
   1344: {1,1,1,1,1,1,2,4}
   1408: {1,1,1,1,1,1,1,5}
   1440: {1,1,1,1,1,2,2,3}
   1600: {1,1,1,1,1,1,3,3}

Counting terms by Heinz weight (in A032766) gives A119620.
An adjoint version is A348550, counted by A108711.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts wiggly permutations of prime factors.
Cf. A000070, A000097, A028982, A236914, A316413, A347457, A348551.

Select[Range[1000],Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==Floor[3*PrimeOmega[#]/2]&]

cp's OEIS Frontend

A100824 Number of partitions of n with at most one odd part.

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A347465 Numbers whose multiset of prime indices has alternating product > 1.

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A347705 Number of factorizations of n with reverse-alternating product > 1.

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A347464 Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1.

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A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

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A363623 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-weighted alternating sum k (leading and trailing 0's omitted).

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A347447 Number of strict factorizations of n with alternating product > 1.

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A347708 Number of distinct possible alternating products of odd-length factorizations of n.

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