A347450 -id:A347450 - cp's OEIS frontend

A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 114, 116, 117, 124, 125, 127, 128, 130

Offset: 1

Gus Wiseman, Sep 24 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 odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.

			The terms and their prime indices begin:
      2: {1}         29: {10}            61: {18}
      3: {2}         31: {11}            63: {2,2,4}
      5: {3}         32: {1,1,1,1,1}     67: {19}
      7: {4}         37: {12}            68: {1,1,7}
      8: {1,1,1}     41: {13}            71: {20}
     11: {5}         42: {1,2,4}         72: {1,1,1,2,2}
     12: {1,1,2}     43: {14}            73: {21}
     13: {6}         44: {1,1,5}         75: {2,3,3}
     17: {7}         45: {2,2,3}         76: {1,1,8}
     18: {1,2,2}     47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     50: {1,3,3}         80: {1,1,1,1,3}
     23: {9}         52: {1,1,6}         83: {23}
     27: {2,2,2}     53: {16}            89: {24}
     28: {1,1,4}     59: {17}            92: {1,1,9}

The reciprocal version is A000290.
Allowing any alternating product <= 1 gives A001105.
Allowing any alternating product gives A026424.
Factorizations of this type are counted by A347441.
These partitions are counted by A347444.
Allowing any length gives A347454.
Allowing any alternating product > 1 gives A347465.
A027193 counts odd-length partitions.
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.
A344606 counts alternating permutations of prime indices.
A347446 counts partitions with integer alternating product.
A347457 ranks partitions with integer alt product, complement A347455.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.

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],OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]

A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 32, 34, 36, 38, 39, 40, 46, 49, 50, 54, 56, 57, 58, 62, 64, 65, 72, 74, 81, 82, 84, 86, 87, 88, 90, 94, 96, 98, 100, 104, 106, 111, 115, 118, 121, 122, 126, 128, 129, 133, 134, 136, 142, 144, 146, 150, 152

Offset: 1

Gus Wiseman, Sep 24 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 reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
Also Heinz numbers integer partitions with integer reverse-reciprocal alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The terms and their prime indices begin:
      1: {}            32: {1,1,1,1,1}       65: {3,6}
      2: {1}           34: {1,7}             72: {1,1,1,2,2}
      4: {1,1}         36: {1,1,2,2}         74: {1,12}
      6: {1,2}         38: {1,8}             81: {2,2,2,2}
      8: {1,1,1}       39: {2,6}             82: {1,13}
      9: {2,2}         40: {1,1,1,3}         84: {1,1,2,4}
     10: {1,3}         46: {1,9}             86: {1,14}
     14: {1,4}         49: {4,4}             87: {2,10}
     16: {1,1,1,1}     50: {1,3,3}           88: {1,1,1,5}
     18: {1,2,2}       54: {1,2,2,2}         90: {1,2,2,3}
     21: {2,4}         56: {1,1,1,4}         94: {1,15}
     22: {1,5}         57: {2,8}             96: {1,1,1,1,1,2}
     24: {1,1,1,2}     58: {1,10}            98: {1,4,4}
     25: {3,3}         62: {1,11}           100: {1,1,3,3}
     26: {1,6}         64: {1,1,1,1,1,1}    104: {1,1,1,6}

The version for reversed prime indices is A028982, counted by A119620.
The additive version is A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609.
Factorizations of this type are counted by A347439.
Allowing any alternating product <= 1 gives A347450.
The non-reciprocal version is A347454.
Allowing any alternating product > 1 gives A347465, reverse A028983.
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.
A344606 counts alternating permutations of prime indices.
A347457 ranks partitions with integer alternating product.
Cf. A001222, A236913, A316523, A344617, A345958, A345959, A346703, A346704, A347437, A347446, A347455.

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[1/altprod[primeMS[#]]]&]

A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256, 264, 270, 280

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than or equal to that of their conjugate.

			The terms together with their prime indices begin:
    1: ()
    2: (1)
    4: (1,1)
    6: (2,1)
    8: (1,1,1)
    9: (2,2)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   20: (3,1,1)
   24: (2,1,1,1)
   27: (2,2,2)
   30: (3,2,1)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   56: (4,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A046682.
The opposite version is A352489, strong A352487.
The strong version is A352490, counted by A000701.
These are the positions of nonnegative terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352525 counts compositions by weak superdiagonals, rank statistic A352517.
Cf. A000720, A045931, A114088, A120383, A175508, A290822, A319005, A325040, A325698, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#>=Times@@Prime/@conj[primeMS[#]]&]

a(n) >= A122111(a(n)).

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&]

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).

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[#]]]]&]

cp's OEIS Frontend

A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

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A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

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A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

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

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

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