A106529 -id:A106529 - cp's OEIS frontend

A340386 Heinz numbers of integer partitions with an odd number of parts, the greatest of which is odd.

2, 5, 8, 11, 17, 20, 23, 30, 31, 32, 41, 44, 45, 47, 50, 59, 66, 67, 68, 73, 75, 80, 83, 92, 97, 99, 102, 103, 109, 110, 120, 124, 125, 127, 128, 137, 138, 149, 153, 154, 157, 164, 165, 167, 170, 176, 179, 180, 186, 188, 191, 197, 200, 207, 211, 227, 230

Offset: 1

Gus Wiseman, Jan 25 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.

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)             59: (17)           120: (3,2,1,1,1)
      5: (3)             66: (5,2,1)        124: (11,1,1)
      8: (1,1,1)         67: (19)           125: (3,3,3)
     11: (5)             68: (7,1,1)        127: (31)
     17: (7)             73: (21)           128: (1,1,1,1,1,1,1)
     20: (3,1,1)         75: (3,3,2)        137: (33)
     23: (9)             80: (3,1,1,1,1)    138: (9,2,1)
     30: (3,2,1)         83: (23)           149: (35)
     31: (11)            92: (9,1,1)        153: (7,2,2)
     32: (1,1,1,1,1)     97: (25)           154: (5,4,1)
     41: (13)            99: (5,2,2)        157: (37)
     44: (5,1,1)        102: (7,2,1)        164: (13,1,1)
     45: (3,2,2)        103: (27)           165: (5,3,2)
     47: (15)           109: (29)           167: (39)
     50: (3,3,1)        110: (5,3,1)        170: (7,3,1)

Note: Heinz numbers are given in parentheses below.
The case of odd length only is A026424.
The case of odd maximum only is A244991.
Positions of odd terms in A326846.
These partitions are counted by A340385.
The version for factorizations is A340607.
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length, or of odd maximum.
A061395 gives maximum prime index.
A106529 lists numbers with Omega equal to maximum prime index.
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
Cf. A001222, A027187, A056239, A112798, A236914, A258116, A300063, A324522, A340608, A340788, A340831.

Select[Range[100],OddQ[PrimeOmega[#]*PrimePi[FactorInteger[#][[-1,1]]]]&]

Intersection of A026424 (odd length) and A244991 (odd maximum).

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

Gus Wiseman, Jan 15 2021

nonn

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

			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.

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.

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

Gus Wiseman, Jan 29 2021

nonn

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

			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)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

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.

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

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

A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 46, 47, 48, 51, 53, 55, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 79, 80, 82, 83, 85, 88, 89, 90, 93, 94, 97, 98, 99

Offset: 1

Gus Wiseman, Jan 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.

			The sequence of terms together with their prime indices begins:
     2: {1}          22: {1,5}          44: {1,1,5}
     3: {2}          23: {9}            46: {1,9}
     4: {1,1}        25: {3,3}          47: {15}
     5: {3}          27: {2,2,2}        48: {1,1,1,1,2}
     7: {4}          28: {1,1,4}        51: {2,7}
     8: {1,1,1}      29: {10}           53: {16}
    10: {1,3}        31: {11}           55: {3,5}
    11: {5}          32: {1,1,1,1,1}    59: {17}
    12: {1,1,2}      33: {2,5}          60: {1,1,2,3}
    13: {6}          34: {1,7}          61: {18}
    15: {2,3}        37: {12}           62: {1,11}
    16: {1,1,1,1}    40: {1,1,1,3}      63: {2,2,4}
    17: {7}          41: {13}           64: {1,1,1,1,1,1}
    18: {1,2,2}      42: {1,2,4}        66: {1,2,5}
    19: {8}          43: {14}           67: {19}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
These are the Heinz numbers of the partitions counted by A200750.
The case of equality is A047993 (A106529).
The divisible instead of coprime version is A168659 (A340609).
The dividing instead of coprime version is A168659 (A340610), with strict case A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A051424 counts singleton or pairwise coprime partitions (A302569).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A259936 counts singleton or pairwise coprime factorizations.
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
A326849 counts partitions whose sum divides length times maximum (A326848).
A327516 counts pairwise coprime partitions (A302696).
Cf. A003114, A039900, A096401, A244990/A244991, A340606, A340653, A340691, A340787/A340788.

Select[Range[100],GCD[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==1&]

A340784 Heinz numbers of even-length integer partitions of even numbers.

Original entry on oeis.org

1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 111, 115, 118, 121, 129, 133, 134, 136, 144, 146, 155, 156, 159, 160, 166, 169, 183, 184, 187, 189, 194, 196, 198, 203, 205, 206, 210, 213, 218, 220

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 28 2024

			The sequence of partitions together with their Heinz numbers begins:
      1: ()            57: (8,2)            118: (17,1)
      4: (1,1)         62: (11,1)           121: (5,5)
      9: (2,2)         64: (1,1,1,1,1,1)    129: (14,2)
     10: (3,1)         81: (2,2,2,2)        133: (8,4)
     16: (1,1,1,1)     82: (13,1)           134: (19,1)
     21: (4,2)         84: (4,2,1,1)        136: (7,1,1,1)
     22: (5,1)         85: (7,3)            144: (2,2,1,1,1,1)
     25: (3,3)         87: (10,2)           146: (21,1)
     34: (7,1)         88: (5,1,1,1)        155: (11,3)
     36: (2,2,1,1)     90: (3,2,2,1)        156: (6,2,1,1)
     39: (6,2)         91: (6,4)            159: (16,2)
     40: (3,1,1,1)     94: (15,1)           160: (3,1,1,1,1,1)
     46: (9,1)        100: (3,3,1,1)        166: (23,1)
     49: (4,4)        111: (12,2)           169: (6,6)
     55: (5,3)        115: (9,3)            183: (18,2)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of prime powers is A056798.
These partitions are counted by A236913.
The odd version is A160786 (A340931).
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
- Even -
A027187 counts partitions of even length/maximum (A028260/A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
A340601 counts partitions of even rank (A340602).
A340785 counts factorizations into even factors.
A340786 counts even-length factorizations into even factors.
Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.
Squares (A000290) is a subsequence.
Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).
Positions of even terms in A373381.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[PrimeOmega[#]]&&EvenQ[Total[primeMS[#]]]&]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
isA340784(n) = A353331(n); \\ Antti Karttunen, Apr 14 2022

Intersection of A028260 and A300061.

A324519 Numbers > 1 where the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

4, 12, 18, 20, 27, 28, 44, 50, 52, 60, 68, 76, 84, 90, 92, 98, 116, 124, 126, 132, 135, 140, 148, 150, 156, 164, 172, 188, 189, 198, 204, 212, 220, 225, 228, 234, 236, 242, 244, 260, 268, 276, 284, 292, 294, 297, 306, 308, 316, 332, 338, 340, 342, 348, 350

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of the integer partitions enumerated by A324520. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   4: {1,1}
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  27: {2,2,2}
  28: {1,1,4}
  44: {1,1,5}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  68: {1,1,7}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A046660, A055396, A056239, A061395, A106529, A112798.

Cf. A324515, A324517, A324520, A324521, A324522, A324560, A324562.

Select[Range[2,100],With[{f=FactorInteger[#]},PrimePi[f[[1,1]]]==Total[Last/@f]-Length[f]]&]

A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

A325233 Heinz numbers of integer partitions with Dyson rank 1.

Original entry on oeis.org

3, 10, 15, 25, 28, 42, 63, 70, 88, 98, 105, 132, 147, 175, 198, 208, 220, 245, 297, 308, 312, 330, 343, 462, 468, 484, 495, 520, 544, 550, 693, 702, 726, 728, 770, 780, 816, 825, 1053, 1078, 1089, 1092, 1144, 1155, 1170, 1210, 1216, 1224, 1300, 1352, 1360

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index is one greater than their number of prime indices counted with multiplicity.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    15: {2,3}
    25: {3,3}
    28: {1,1,4}
    42: {1,2,4}
    63: {2,2,4}
    70: {1,3,4}
    88: {1,1,1,5}
    98: {1,4,4}
   105: {2,3,4}
   132: {1,1,2,5}
   147: {2,4,4}
   175: {3,3,4}
   198: {1,2,2,5}
   208: {1,1,1,1,6}
   220: {1,1,3,5}
   245: {3,4,4}
   297: {2,2,2,5}
   308: {1,1,4,5}

FindStat, St000145: The Dyson rank of a partition

Positions of 1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325234, A325235.

Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==1&]

A326845 Sum times maximum of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 4, 2, 9, 6, 16, 3, 8, 12, 25, 8, 36, 20, 15, 4, 49, 10, 64, 15, 24, 30, 81, 10, 18, 42, 12, 24, 100, 18, 121, 5, 35, 56, 28, 12, 144, 72, 48, 18, 169, 28, 196, 35, 21, 90, 225, 12, 32, 21, 63, 48, 256, 14, 40, 28, 80, 110, 289, 21, 324, 132, 32, 6, 54, 40

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Cf. A056239, A061395, A106529, A112798, A316413, A326836, A326844, A326846.

Table[If[n==1,0,With[{y=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[y]*Max[y]]],{n,100}]

a(n) = A056239(n) * A061395(n).

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

Gus Wiseman, Jan 16 2021

nonn

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).

			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.

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.

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

Gus Wiseman, Jan 17 2021

nonn

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).

			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.

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.

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/@#]&]!={}&]

cp's OEIS Frontend

A340386 Heinz numbers of integer partitions with an odd number of parts, the greatest of which is odd.

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A340597 Numbers with an alt-balanced factorization.

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A340692 Number of integer partitions of n of odd rank.

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A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

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A340784 Heinz numbers of even-length integer partitions of even numbers.

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A324519 Numbers > 1 where the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

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A325233 Heinz numbers of integer partitions with Dyson rank 1.

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A326845 Sum times maximum of the integer partition with Heinz number n.

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