A071322 -id:A071322 - cp's OEIS frontend

A344616 Alternating sum of the integer partition with Heinz number n.

0, 1, 2, 0, 3, 1, 4, 1, 0, 2, 5, 2, 6, 3, 1, 0, 7, 1, 8, 3, 2, 4, 9, 1, 0, 5, 2, 4, 10, 2, 11, 1, 3, 6, 1, 0, 12, 7, 4, 2, 13, 3, 14, 5, 3, 8, 15, 2, 0, 1, 5, 6, 16, 1, 2, 3, 6, 9, 17, 1, 18, 10, 4, 0, 3, 4, 19, 7, 7, 2, 20, 1, 21, 11, 2, 8, 1, 5, 22, 3, 0, 12

Offset: 1

Gus Wiseman, Jun 03 2021

nonn

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Also the reverse-alternating sum of the prime indices of n.

			The partition (6,4,3,2,2) has Heinz number 4095 and conjugate (5,5,3,2,1,1), so a(4095) = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Positions of nonzeros are A000037.
Positions of 0's are A000290.
The version for prime factors is A071321 (reverse: A071322).
A version for compositions is A124754.
The version for prime multiplicities is A316523.
The reverse version is A316524, with sign A344617.
A000041 counts partitions of 2n with alternating sum 0.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum.
A335433 ranks separable partitions.
A335448 ranks inseparable partitions.
A344606 counts wiggly permutations of prime indices with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344612 counts partitions by sum and reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A001222, A026424, A028260, A116406, A119899, A343938, A344607, A344608, A344609, A344619, A344653, A344739.

a:= n-> (l-> -add(l[i]*(-1)^i, i=1..nops(l)))(sort(map(
    i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..82);  # Alois P. Heinz, Jun 04 2021

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[ats[Reverse[primeMS[n]]],{n,100}]

a(n) = A257991(A122111(n)).

A057427(a(n)) = A049240(n).

A344607 Number of integer partitions of n with reverse-alternating sum >= 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 15, 16, 27, 29, 48, 52, 81, 90, 135, 151, 220, 248, 352, 400, 553, 632, 859, 985, 1313, 1512, 1986, 2291, 2969, 3431, 4394, 5084, 6439, 7456, 9357, 10836, 13479, 15613, 19273, 22316, 27353, 31659, 38558, 44601, 53998, 62416, 75168

Offset: 0

Gus Wiseman, May 29 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed integer partitions of n with alternating sum >= 0.
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd.
All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.
Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (321)     (421)      (422)
                                     (411)     (511)      (431)
                                     (2211)    (22111)    (521)
                                     (21111)   (31111)    (611)
                                     (111111)  (1111111)  (2222)
                                                          (3311)
                                                          (22211)
                                                          (32111)
                                                          (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

The non-reversed version is A000041.
The opposite version (rev-alt sum <= 0) is A027187, ranked by A028260.
The strict case for n > 0 is A067659 (even bisection: A344650).
The ordered version appears to be A116406 (even bisection: A114121).
The odd bisection is A160786.
The complement is counted by A344608.
The Heinz numbers of these partitions are A344609 (complement: A119899).
The even bisection is A344611.
A000070 counts partitions with alternating sum 1 (reversed: A000004).
A000097 counts partitions with alternating sum 2 (reversed: A120452).
A035363 counts partitions with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum.
A316524 is the alternating sum of prime indices of n (reversed: A344616).
A325534/A325535 count separable/inseparable partitions.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344612 counts partitions by sum and reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A006330, A071321, A071322, A124754, A239829, A239830, A344604, A344651, A344654, A344739, A344742.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30}]

a(n) + A344608(n) = A000041(n).

a(2n+1) = A160786(n).

A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 28 2021

nonn

Differs from A335448 in having a(x^2) = 0 and a(270) = 0.
These are permutations of the prime factors of n, counting multiplicity, with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins (x,x) is A345164, which is identical to this sequence except when n is the square of a prime.

			The permutations for n = 2, 6, 30, 180, 210, 300, 420, 720, 840:
  2   23   253   23253   2537   25253   23275   2323252   232527
      32   325   32325   2735   25352   25273   2325232   232725
           352   32523   3275   32525   25372   2523232   252327
           523   35232   3527   35252   27253             252723
                 52323   3725   52325   27352             272325
                         5273   52523   32527             272523
                         5372           32725             325272
                         5723           35272             327252
                         7253           37252             523272
                         7352           52327             527232
                                        52723             723252
                                        57232             725232
                                        72325
                                        72523
For example, there are no alternating permutations of the prime factors of 270 because the only anti-runs are {3,2,3,5,3} and {3,5,3,2,3}, neither of which is alternating, so a(270) = 0.

The version for permutations is A001250.
The extension to anti-run permutations is A335452.
The version for compositions is A344604.
The version for patterns is A344605.
Positions of zeros are A344653 (counted by A344654).
Not including twins (x,x) gives A345164.
A008480 counts permutations of prime indices (strict: A335489, rank: A333221).
A056239 adds up prime indices,  row sums of A112798.
A071321 and A071322 are signed sums of prime factors.
A316523 is a signed sum of prime multiplicities.
A316524 and A344616 are signed sums of prime indices.
A325534 counts separable partitions (ranked by A335433).
A325535 counts inseparable partitions (ranked by A335448).
A344740 counts partitions with an alternating permutation or twin (x,x).
Cf. A000041, A000607, A000961, A001222, A003242, A026424, A028260, A049774, A103919, A181796, A343938, A344652.

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,100}]

A344653 Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 125, 128, 135, 136, 144, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378, 384

Offset: 1

Gus Wiseman, Jun 12 2021

nonn

Differs from A335448 in lacking squares and having 270 etc.
First differs from A345193 in having 270.
Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
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 of terms together with their prime indices begins:
   8: {1,1,1}
  16: {1,1,1,1}
  24: {1,1,1,2}
  27: {2,2,2}
  32: {1,1,1,1,1}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  80: {1,1,1,1,3}
  81: {2,2,2,2}
  88: {1,1,1,5}
  96: {1,1,1,1,1,2}
For example, 36 has prime indices (1,1,2,2), which has the two wiggly permutations (1,2,1,2) and (2,1,2,1), so 36 is not in the sequence.

A superset of A335448, counted by A325535.
Positions of 0's in A344606.
These partitions are counted by A344654.
The complement is A344742, counted by A344740.
The separable case is A345173, counted by A345166.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A344604 counts wiggly compositions with twins.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344614, A344615, A344616, A344652, A345163, A345168, A345193.

Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]=={}&]

Complement of A001248 in A345171.

A345172 Numbers whose multiset of prime factors has an alternating permutation.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A212167 in containing 72.
First differs from A335433 in lacking 270, corresponding to the partition (3,2,2,2,1).
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
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 of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

Including squares of primes A001248 gives A344742, counted by A344740.
This is a subset of A335433, which is counted by A325534.
Positions of nonzero terms in A345164.
The partitions with these Heinz numbers are counted by A345170.
The complement is A345171, which is counted by A345165.
A345173 = A345171 /\ A335433 is counted by A345166.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344606 counts alternating permutations of prime indices with twins.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344605, A344614, A344616, A344653, A344654, A345163, A345167, A347706, A348379.

wigQ[y_]:=Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1;
Select[Range[100],Select[Permutations[ Flatten[ConstantArray@@@FactorInteger[#]]],wigQ[#]&]!={}&]

Complement of A001248 (squares of primes) in A344742.

A345164 Number of alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.

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 permutation, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
  (12132)  (1324)  (13132)  (12143)  (121323)
  (21213)  (1423)  (13231)  (13142)  (132312)
  (21312)  (2143)  (21313)  (13241)  (213132)
  (23121)  (2314)  (23131)  (14132)  (213231)
  (31212)  (2413)  (31213)  (14231)  (231213)
           (3142)  (31312)  (21314)  (231312)
           (3241)           (21413)  (312132)
           (3412)           (23141)  (323121)
           (4132)           (24131)
           (4231)           (31214)
                            (31412)
                            (34121)
                            (41213)
                            (41312)

Counting all permutations gives A008480.
Dominated by A335452 (number of separations of prime factors).
Including twins (x,x) gives A344606.
Positions of zeros are A345171, counted by A345165.
Positions of nonzero terms are A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.
A344740 counts twins and partitions w/ alternating permutation, rank: A344742.
A345166 counts separable partitions w/o alternating permutation, rank: A345173.
A345170 counts partitions with a alternating permutation.
Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],wigQ]],{n,30}]

A347437 Number of factorizations of n with integer alternating product.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 6, 1, 1, 1, 2, 1, 1, 1, 7

Offset: 1

Gus Wiseman, Sep 06 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 factorizations for n = 4, 16, 36, 48, 54, 64, 108:
  (4)   (16)      (36)      (48)        (54)    (64)          (108)
  (2*2) (4*4)     (6*6)     (2*4*6)     (2*3*9) (8*8)         (2*6*9)
        (2*2*4)   (2*2*9)   (3*4*4)     (3*3*6) (2*4*8)       (3*6*6)
        (2*2*2*2) (2*3*6)   (2*2*12)            (4*4*4)       (2*2*27)
                  (3*3*4)   (2*2*2*2*3)         (2*2*16)      (2*3*18)
                  (2*2*3*3)                     (2*2*4*4)     (3*3*12)
                                                (2*2*2*2*4)   (2*2*3*3*3)
                                                (2*2*2*2*2*2)

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

Positions of 1's are A005117, complement A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The restriction to powers of 2 is A344607.
The even-length case is A347438, also the case of alternating product 1.
The reciprocal version is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441.
The reverse version is A347442.
The additive version is A347446, ranked by A347457.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347458, reciprocal A347459.
The ordered version is A347463.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors of n (reverse: A071322).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.

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],IntegerQ@*altprod]],{n,100}]

A347437(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)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A344607(n).

a(n^2) = A347458(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

A071321 Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n).

Original entry on oeis.org

0, 2, 3, 0, 5, -1, 7, 2, 0, -3, 11, 3, 13, -5, -2, 0, 17, 2, 19, 5, -4, -9, 23, -1, 0, -11, 3, 7, 29, 4, 31, 2, -8, -15, -2, 0, 37, -17, -10, -3, 41, 6, 43, 11, 5, -21, 47, 3, 0, 2, -14, 13, 53, -1, -6, -5, -16, -27, 59, -2, 61, -29, 7, 0, -8

Offset: 1

Reinhard Zumkeller, May 18 2002

sign

a(n) = 0 iff n square, a(A000290(n)) = 0;
a(n) <= 0 iff A001222(n) is even;
a(n) = n iff n prime, a(A000040(n)) = A000040(n).
a(2n) = -a(n) + 2. - Ralf Stephan

			72 = 2*2*2*3*3, therefore a(72) = 2 - 2 + 2 - 3 + 3 = 2;
90 = 2*3*3*5, therefore a(90) = 2 - 3 + 3 - 5 = -3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001414, A020639, A071322, A071323.
Cf. A008836.
Cf. A027746, A033999.

a071321 1 = 0
a071321 n = sum $ zipWith (*) a033999_list $ a027746_row n
-- Reinhard Zumkeller, Jun 01 2013

Join[{0},Table[Total[Times@@@Partition[Riffle[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[n]],{1,-1},{2,-1,2}],2]],{n,2,100}]] (* Harvey P. Dale, Sep 23 2015 *)

from sympy import factorint
def A071321(n):
    fs = factorint(n,multiple=True)
    return sum(fs[::2])-sum(fs[1::2]) # Chai Wah Wu, Aug 23 2021

a(n) = -A071322(n)*A008836(n). - Franklin T. Adams-Watters, Oct 18 2006

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 7, 14, 15, 27, 29, 49, 54, 86, 96, 146, 165, 242, 275, 392, 449, 623, 716, 973, 1123, 1498, 1732, 2274, 2635, 3411, 3955, 5059, 5871, 7427, 8620, 10801, 12536, 15572, 18065, 22267, 25821, 31602, 36617, 44533, 51560, 62338, 72105, 86716

Offset: 0

Gus Wiseman, May 30 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed of integer partitions of n with alternating sum < 0.
No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)    (42)    (43)      (53)      (54)
              (41)    (51)    (52)      (62)      (63)
              (2111)  (3111)  (61)      (71)      (72)
                              (2221)    (3221)    (81)
                              (3211)    (4211)    (3222)
                              (4111)    (5111)    (3321)
                              (211111)  (311111)  (4221)
                                                  (4311)
                                                  (5211)
                                                  (6111)
                                                  (222111)
                                                  (321111)
                                                  (411111)
                                                  (21111111)

The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.
The strict case (for n > 2) is A067659 (odd bisection: A344650).
The Heinz numbers of these partitions are A119899 (complement: A344609).
The bisections are A236914 (odd) and A344743 (even).
The ordered version appears to be A294175 (even bisection: A008549).
The complement is counted by A344607 (even bisection: A344611).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with alternating sum <= 0, ranked by A028260.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions with rev-alternating sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30}]

A347438 Number of unordered factorizations of n with alternating product 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 06 2021

nonn

Also the number of unordered factorizations of n with alternating sum 0.
Also the number of unordered factorizations of n with all even multiplicities.
This is the even-length case of A347437, the odd-length case being A347441.
An unordered 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 a(n) factorizations for n = 16, 64, 144, 256, 576:
  4*4      8*8          12*12        16*16            24*24
  2*2*2*2  2*2*4*4      2*2*6*6      2*2*8*8          3*3*8*8
           2*2*2*2*2*2  3*3*4*4      4*4*4*4          4*4*6*6
                        2*2*2*2*3*3  2*2*2*2*4*4      2*2*12*12
                                     2*2*2*2*2*2*2*2  2*2*2*2*6*6
                                                      2*2*3*3*4*4
                                                      2*2*2*2*2*2*3*3

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

Positions of zeros are A000037.
Positions of nonzero terms are A000290.
The restriction to perfect squares is A001055 (ordered: A273013).
The restriction to powers of 2 is A035363.
The additive version is A119620, ranked by A028982.
Positions of non-1's are A213367 \ {1}.
Positions of 1's are A280076 = {1} \/ A001248.
Sorted first positions are 1, 2, and all terms of A330972 squared.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
Allowing any integer alternating product gives A347437.
Allowing any integer reciprocal alternating product gives A347439.
Allowing any alternating product < 1 gives A347440.
Allowing any alternating product >= 1 gives A347456.
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).
A344606 counts alternating permutations of prime factors.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, 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],altprod[#]==1&]],{n,100}]

A347438(n, m=n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A347438(n/d, d, 1-k, t*(d^((-1)^k))))); (s)); \\ Antti Karttunen, Oct 30 2021

a(2^n) = A035363(n).

a(n^2) = A001055(n).

Name and comments clarified (with unordered) by Jacob Sprittulla, Oct 05 2021

cp's OEIS Frontend

A344616 Alternating sum of the integer partition with Heinz number n.

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A344607 Number of integer partitions of n with reverse-alternating sum >= 0.

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A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

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A344653 Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A345172 Numbers whose multiset of prime factors has an alternating permutation.

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A345164 Number of alternating permutations of the multiset of prime factors of n.

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A347437 Number of factorizations of n with integer alternating product.

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A071321 Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n).

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