cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A340655 Number of twice-balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 2, 0, 2, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Jan 15 2021

Keywords

Comments

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

Examples

			The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
  2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
  3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
        2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
        2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
                 3*4*30    4*8*15    4*5*45    3*3*5*56
                 3*6*20    5*8*12    5*6*30    3*3*7*40
                 3*8*15    6*8*10    5*9*20    3*5*7*24
                 4*5*18    2*12*20   2*10*45   2*2*2*315
                 5*6*12    4*10*12   2*15*30   2*2*3*210
                 2*10*18             2*18*25   2*2*5*126
                 2*12*15             3*10*30   2*3*3*140
                 3*10*12             3*12*25
                                     3*15*20
                                     5*10*18
                                     5*12*15

Crossrefs

The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340652.
The balanced version is A340653.
The cross-balanced version is A340654.
Positions of zeros are A340656.
Positions of nonzero terms are A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, A340608.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],#=={}||Length[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]

A340597 Numbers with an alt-balanced factorization.

Original entry on oeis.org

4, 12, 18, 27, 32, 48, 64, 72, 80, 96, 108, 120, 128, 144, 160, 180, 192, 200, 240, 256, 270, 288, 300, 320, 360, 384, 400, 405, 432, 448, 450, 480, 500, 540, 576, 600, 640, 648, 672, 675, 720, 750, 768, 800, 864, 896, 900, 960, 972, 1000, 1008, 1024, 1080
Offset: 1

Views

Author

Gus Wiseman, Jan 15 2021

Keywords

Comments

We define a factorization into factors > 1 to be alt-balanced if its length is equal to its greatest factor.

Examples

			The sequence of terms together with their prime signatures begins:
      4: (2)        180: (2,2,1)    450: (1,2,2)
     12: (2,1)      192: (6,1)      480: (5,1,1)
     18: (1,2)      200: (3,2)      500: (2,3)
     27: (3)        240: (4,1,1)    540: (2,3,1)
     32: (5)        256: (8)        576: (6,2)
     48: (4,1)      270: (1,3,1)    600: (3,1,2)
     64: (6)        288: (5,2)      640: (7,1)
     72: (3,2)      300: (2,1,2)    648: (3,4)
     80: (4,1)      320: (6,1)      672: (5,1,1)
     96: (5,1)      360: (3,2,1)    675: (3,2)
    108: (2,3)      384: (7,1)      720: (4,2,1)
    120: (3,1,1)    400: (4,2)      750: (1,1,3)
    128: (7)        405: (4,1)      768: (8,1)
    144: (4,2)      432: (4,3)      800: (5,2)
    160: (5,1)      448: (6,1)      864: (5,3)
For example, there are two alt-balanced factorizations of 480, namely (2*3*4*4*5) and (2*2*2*2*5*6), so 480 in the sequence.

Crossrefs

Numbers with a balanced factorization are A100959.
These factorizations are counted by A340599.
The twice-balanced version is A340657.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
- A340655 counts twice-balanced factorizations.
Cf. A006141, A064174, A117409, A200750, A303975, A324518, A324522, A325134, A340607, A340608, A340611, A340656.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],Length[#]==Max[#]&]!={}&]

A340692 Number of integer partitions of n of odd rank.

Original entry on oeis.org

0, 0, 2, 0, 4, 2, 8, 4, 14, 12, 26, 22, 44, 44, 76, 78, 126, 138, 206, 228, 330, 378, 524, 602, 814, 950, 1252, 1466, 1900, 2238, 2854, 3362, 4236, 5006, 6232, 7356, 9078, 10720, 13118, 15470, 18800, 22152, 26744, 31456, 37772, 44368, 53002, 62134, 73894
Offset: 0

Views

Author

Gus Wiseman, Jan 29 2021

Keywords

Comments

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots):
  .  .  (2)   .  (4)     (32)   (6)       (52)     (8)         (54)
        (11)     (31)    (221)  (33)      (421)    (53)        (72)
                 (211)          (51)      (3211)   (71)        (432)
                 (1111)         (222)     (22111)  (422)       (441)
                                (411)              (431)       (621)
                                (3111)             (611)       (3222)
                                (21111)            (3221)      (3321)
                                (111111)           (3311)      (5211)
                                                   (5111)      (22221)
                                                   (22211)     (42111)
                                                   (41111)     (321111)
                                                   (311111)    (2211111)
                                                   (2111111)
                                                   (11111111)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of length/maximum instead of rank is A027193 (A026424/A244991).
The case of odd positive rank is A101707 is (A340604).
The strict case is A117193.
The even version is A340601 (A340602).
The Heinz numbers of these partitions are (A340603).
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064173 counts partitions of positive/negative rank (A340787/A340788).
A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
A101198 counts partitions of rank 1 (A325233).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A003114, A006141, A027187, A039900, A067538, A096401, A117409, A143773, A324518, A325134, A340828, A340854/A340855.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],OddQ[Max[#]-Length[#]]&]],{n,0,30}]

Formula

Having odd rank is preserved under conjugation, and self-conjugate partitions cannot have odd rank, so a(n) = 2*A101707(n) for n > 0.

A340656 Numbers without a twice-balanced factorization.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 42, 46, 48, 49, 51, 55, 57, 58, 60, 62, 64, 65, 66, 69, 70, 72, 74, 77, 78, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 102, 105, 106, 108, 110, 111, 112, 114, 115, 118, 119
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2021

Keywords

Comments

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

Examples

			The sequence of terms together with their prime indices begins:
     4: {1,1}          33: {2,5}          64: {1,1,1,1,1,1}
     6: {1,2}          34: {1,7}          65: {3,6}
     8: {1,1,1}        35: {3,4}          66: {1,2,5}
     9: {2,2}          38: {1,8}          69: {2,9}
    10: {1,3}          39: {2,6}          70: {1,3,4}
    14: {1,4}          42: {1,2,4}        72: {1,1,1,2,2}
    15: {2,3}          46: {1,9}          74: {1,12}
    16: {1,1,1,1}      48: {1,1,1,1,2}    77: {4,5}
    21: {2,4}          49: {4,4}          78: {1,2,6}
    22: {1,5}          51: {2,7}          80: {1,1,1,1,3}
    25: {3,3}          55: {3,5}          81: {2,2,2,2}
    26: {1,6}          57: {2,8}          82: {1,13}
    27: {2,2,2}        58: {1,10}         84: {1,1,2,4}
    30: {1,2,3}        60: {1,1,2,3}      85: {3,7}
    32: {1,1,1,1,1}    62: {1,11}         86: {1,14}
For example, the factorizations of 48 with (2) and (3) equal are: (2*2*2*6), (2*2*3*4), (2*4*6), (3*4*4), but since none of these has length 2, the sequence contains 48.

Crossrefs

Positions of zeros in A340655.
The complement is A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
Cf. A112798, A117409, A325134, A339846, A339890, A340607, A340689, A340690.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],#=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]=={}&]

A340657 Numbers with a twice-balanced factorization.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 36, 37, 40, 41, 43, 44, 45, 47, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 88, 89, 92, 97, 98, 99, 100, 101, 103, 104, 107, 109, 113, 116, 117, 120, 124, 127, 131, 135, 136, 137
Offset: 1

Views

Author

Gus Wiseman, Jan 17 2021

Keywords

Comments

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            29: {10}          59: {17}
      2: {1}           31: {11}          61: {18}
      3: {2}           36: {1,1,2,2}     63: {2,2,4}
      5: {3}           37: {12}          67: {19}
      7: {4}           40: {1,1,1,3}     68: {1,1,7}
     11: {5}           41: {13}          71: {20}
     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}          79: {22}
     19: {8}           50: {1,3,3}       83: {23}
     20: {1,1,3}       52: {1,1,6}       88: {1,1,1,5}
     23: {9}           53: {16}          89: {24}
     24: {1,1,1,2}     54: {1,2,2,2}     92: {1,1,9}
     28: {1,1,4}       56: {1,1,1,4}     97: {25}
The twice-balanced factorizations of 1920 (with prime indices {1,1,1,1,1,1,1,2,3}) are (8*8*30) and (8*12*20), so 1920 is in the sequence.

Crossrefs

The alt-balanced version is A340597.
Positions of nonzero terms in A340655.
The complement is A340656.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
Cf. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],#=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]!={}&]

A340788 Heinz numbers of integer partitions of negative rank.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405
Offset: 1

Views

Author

Gus Wiseman, Jan 29 2021

Keywords

Comments

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.
The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      4: (1,1)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 is (A340929).
The even case is A101708 is (A340930).
The positive version is (A340787).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]

Formula

For all terms A061395(a(n)) < A001222(a(n)).

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Gus Wiseman, Jan 29 2021

Keywords

Comments

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.
The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The sequence of partitions together with their Heinz numbers begins:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>PrimeOmega[#]&]

Formula

For all terms A061395(a(n)) > A001222(a(n)).

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2021

Keywords

Examples

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]

A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350
Offset: 0

Views

Author

Gus Wiseman, Jan 28 2021

Keywords

Comments

Also the number of integer partitions of n with maximum 2^k where k is the length.

Examples

			The partitions for n = 12, 14, 16, 22, 24:
  32211111  32222111  32222221  33333322          33333333
  33111111  33221111  33222211  33333331          4222221111111111
            33311111  33322111  4222111111111111  4322211111111111
                      33331111  4321111111111111  4332111111111111
                                4411111111111111  4422111111111111
                                                  4431111111111111
The conjugate partitions:
  (8,2,2)  (8,3,3)  (8,4,4)  (8,7,7)     (8,8,8)
  (8,3,1)  (8,4,2)  (8,5,3)  (8,8,6)     (16,3,3,2)
           (8,5,1)  (8,6,2)  (16,2,2,2)  (16,4,2,2)
                    (8,7,1)  (16,3,2,1)  (16,4,3,1)
                             (16,4,1,1)  (16,5,2,1)
                                         (16,6,1,1)

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
A340689 have a factorization of length 2^max.
A340690 have a factorization of maximum 2^length.
Cf. A003114, A006141, A064174, A098124, A117409, A200750, A340385, A340387, A340599, A340601.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]

A340689 Numbers with a factorization of length 2^k into factors > 1, where k is the greatest factor.

Original entry on oeis.org

1, 16, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 131072, 196608, 262144, 294912, 393216, 442368, 524288, 589824, 663552, 786432, 884736, 995328, 1048576, 1179648, 1327104, 1492992, 1572864, 1769472, 1990656, 2097152, 2239488, 2359296, 2654208, 2985984, 3145728
Offset: 1

Views

Author

Gus Wiseman, Jan 28 2021

Keywords

Examples

			The initial terms and a valid factorization of each are:
         1 =
        16 = 2*2*2*2
       384 = 2*2*2*2*2*2*2*3
       576 = 2*2*2*2*2*2*3*3
       864 = 2*2*2*2*2*3*3*3
      1296 = 2*2*2*2*3*3*3*3
      1944 = 2*2*2*3*3*3*3*3
      2916 = 2*2*3*3*3*3*3*3
      4374 = 2*3*3*3*3*3*3*3
      6561 = 3*3*3*3*3*3*3*3
    131072 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*4
    196608 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*3*4
    262144 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*4*4
    294912 = 2*2*2*2*2*2*2*2*2*2*2*2*2*3*3*4

Links

Crossrefs

Partitions of the prescribed type are counted by A340611.
The conjugate version is A340690.
A001055 counts factorizations, with strict case A045778.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],Length[#]==2^Max@@#&]!={}&]

Extensions

More terms from Chai Wah Wu, Feb 01 2021
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