A347462 -id:A347462 - cp's OEIS frontend

A347444 Number of odd-length integer partitions of n with integer alternating product.

0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398

Offset: 0

Gus Wiseman, Sep 14 2021

nonn

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

The reverse version (integer reverse-alternating product) is the same.

			The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)        (9)
            (111)  (211)  (221)    (222)    (322)      (332)      (333)
                          (311)    (411)    (331)      (422)      (441)
                          (11111)  (21111)  (421)      (611)      (522)
                                            (511)      (22211)    (621)
                                            (22111)    (41111)    (711)
                                            (31111)    (2111111)  (22221)
                                            (1111111)             (32211)
                                                                  (33111)
                                                                  (42111)
                                                                  (51111)
                                                                  (2211111)
                                                                  (3111111)
                                                                  (111111111)

The reciprocal version is A035363.
Allowing any alternating product gives A027193.
The multiplicative version (factorizations) is A347441.
Allowing any length gives A347446, reverse A347445.
Allowing any length and alternating product > 1 gives A347448.
Allowing any reverse-alternating product > 1 gives A347449.
Ranked by A347453.
The even-length instead of odd-length version is A347704.
A000041 counts partitions.
A000302 counts odd-length compositions, ranked by A053738.
A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A027187 counts partitions of even length, strict A067661.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.

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

A347449 Number of integer partitions of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 10, 11, 20, 22, 37, 41, 66, 75, 113, 129, 190, 218, 310, 358, 497, 576, 782, 908, 1212, 1411, 1851, 2156, 2793, 3255, 4163, 4853, 6142, 7159, 8972, 10451, 12989, 15123, 18646, 21689, 26561, 30867, 37556, 43599, 52743, 61161, 73593

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

All such partitions have odd length.

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(2) = 1 through a(9) = 11 partitions:
  (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)
            (211)  (311)  (222)    (322)    (332)      (333)
                          (321)    (421)    (422)      (432)
                          (411)    (511)    (431)      (522)
                          (21111)  (31111)  (521)      (531)
                                            (611)      (621)
                                            (22211)    (711)
                                            (32111)    (32211)
                                            (41111)    (42111)
                                            (2111111)  (51111)
                                                       (3111111)

The strict case is A067659, except that a(0) = a(1) = 0.
The even bisection is A236559.
The non-reverse multiplicative version is A339890, weak A347456.
The case of >= 1 instead of > 1 is A344607.
The opposite version is A344608, also the non-reverse even-length case.
The complement is counted by A347443, non-reverse A119620.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
Reverse version of A347448; also the odd-length case.
The Heinz numbers of these partitions are the complement of A347450.
The multiplicative version (factorizations) is A347705.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A100824 counts partitions of n with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A008549, A086543, A182616, A236913, A325534, A325535, A344611, A347442, A347444, A347447, A347453, A347461, A347465.

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

a(n) = A344607(n) - A119620(n).

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&]

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}]

A347455 Heinz numbers of integer partitions with non-integer alternating product.

Original entry on oeis.org

15, 30, 33, 35, 51, 55, 60, 66, 69, 70, 77, 85, 91, 93, 95, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 154, 155, 161, 165, 170, 177, 182, 186, 187, 190, 201, 203, 204, 205, 209, 210, 215, 217, 219, 220, 221, 231, 238, 240, 246, 247, 249

Offset: 1

Gus Wiseman, Oct 04 2021

nonn

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.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has non-integer reverse-alternating product.

			The terms and their reversed prime indices begin:
     15: (3,2)        102: (7,2,1)        161: (9,4)
     30: (3,2,1)      105: (4,3,2)        165: (5,3,2)
     33: (5,2)        110: (5,3,1)        170: (7,3,1)
     35: (4,3)        119: (7,4)          177: (17,2)
     51: (7,2)        120: (3,2,1,1,1)    182: (6,4,1)
     55: (5,3)        123: (13,2)         186: (11,2,1)
     60: (3,2,1,1)    132: (5,2,1,1)      187: (7,5)
     66: (5,2,1)      135: (3,2,2,2)      190: (8,3,1)
     69: (9,2)        138: (9,2,1)        201: (19,2)
     70: (4,3,1)      140: (4,3,1,1)      203: (10,4)
     77: (5,4)        141: (15,2)         204: (7,2,1,1)
     85: (7,3)        143: (6,5)          205: (13,3)
     91: (6,4)        145: (10,3)         209: (8,5)
     93: (11,2)       154: (5,4,1)        210: (4,3,2,1)
     95: (8,3)        155: (11,3)         215: (14,3)
For example, (4,3,2,1) has alternating product 4/3*2/1 = 8/3, so the Heinz number 210 is in the sequence.

The reciprocal version is A028983, complement A028982.
Factorizations not of this type are counted by A347437.
Partitions not of this type are counted by A347446.
The complement of the reverse reciprocal version is A347451.
The complement in the odd-length case is A347453.
The complement of the reverse version is A347454.
The complement is A347457.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A347461 counts possible alternating products of partitions, reverse A347462.
Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450, A347465.

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],!IntegerQ[altprod[Reverse[primeMS[#]]]]&]

A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 12, 14, 14, 15, 15, 18, 17, 19, 18, 20, 20, 22, 21, 25, 23, 26, 25, 28, 26, 29, 27, 31, 29, 32, 31, 34, 33, 35, 34, 38, 35, 41, 37, 42, 40, 43, 41, 45, 42, 46, 44, 48, 45, 50, 46, 52, 49, 53

Offset: 0

Gus Wiseman, Oct 13 2021

nonn

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.

			Representative partitions for each of the a(16) = 11 alternating products:
         (16) -> 16
     (14,1,1) -> 14
     (12,2,2) -> 12
     (10,3,3) -> 10
      (8,4,4) -> 8
  (9,3,2,1,1) -> 6
     (10,4,2) -> 5
     (12,3,1) -> 4
  (6,4,2,2,2) -> 3
     (10,5,1) -> 2
        (8,8) -> 1

The even-length version is A000035.
The non-reverse version is A028310.
The version for factorizations has special cases:
- no changes: A046951
- non-reverse: A046951
- non-integer: A038548
- odd-length: A046951 + A010052
- non-reverse non-integer: A347460
- non-integer odd-length: A347708
- non-reverse odd-length: A046951 + A010052
- non-reverse non-integer odd-length: A347708
The odd-length version is a(n) - A059841(n).
These partitions are counted by A347445, non-reverse A347446.
Counting non-integers gives A347462, non-reverse A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A119620 counts partitions with alternating product 1, ranked by A028982.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A304792 counts distinct subset-sums of partitions.
A325534 counts separable partitions, complement A325535.
A345926 counts possible alternating sums of permutations of prime indices.
Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[Union[revaltprod/@IntegerPartitions[n]],IntegerQ]],{n,0,30}]

cp's OEIS Frontend

A347444 Number of odd-length integer partitions of n with integer alternating product.

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A347449 Number of integer partitions of n with reverse-alternating product > 1.

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

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

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

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A347455 Heinz numbers of integer partitions with non-integer alternating product.

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A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

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