A324522 -id:A324522 - cp's OEIS frontend

A340607 Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 2, 0, 1, 3, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 2, 0, 1, 4

Offset: 1

Gus Wiseman, Jan 25 2021

nonn

			The a(n) factorizations for n = 27, 84, 108, 180, 252, 360, 432:
  27     2*6*7   2*6*9      4*5*9      4*7*9      5*8*9       6*8*9
  3*3*3  3*4*7   3*4*9      2*2*45     6*6*7      2*4*45      2*8*27
         2*2*21  2*2*27     2*6*15     2*2*63     3*8*15      4*4*27
                 2*2*3*3*3  3*4*15     2*6*21     4*6*15      2*2*2*6*9
                            2*2*3*3*5  3*4*21     2*12*15     2*2*3*4*9
                                       2*2*3*3*7  2*2*2*5*9   2*2*2*2*27
                                                  2*3*3*4*5   2*2*2*2*3*3*3
                                                  2*2*2*3*15

Antti Karttunen, Table of n, a(n) for n = 1..20000

Note: Heinz numbers are given in parentheses below.
The case of odd length only is A339890.
The case of all odd factors is A340102.
The version for partitions is A340385.
The version for prime indices is A340386.
The case of odd maximum only is A340831.
A000009 counts partitions into odd parts (A066208).
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A078408 counts odd-length partitions into odd numbers (A300272).
A316439 counts factorizations by sum and length.
A340101 counts factorizations (into odd factors = of odd numbers).
A340832 counts factorizations whose least part is odd.
A340854/A340855 lack/have a factorization with odd minimum.
Cf. A000700, A024429, A026804, A028260, A061395, A112798, A160786, A236914, A324522, A326845, A340608, A340788.

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],OddQ[Length[#]]&&OddQ[Max@@#]&]],{n,100}]

A340607(n, m=n, k=0, grodd=0) = if(1==n, k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(grodd||(d%2)), s += A340607(n/d, d, 1-k, bitor(1,grodd)))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

A324517 Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

4, 24, 27, 36, 54, 80, 200, 224, 240, 360, 405, 500, 540, 600, 625, 672, 675, 704, 784, 810, 900, 1008, 1120, 1125, 1250, 1350, 1500, 1512, 1664, 1701, 1875, 2112, 2250, 2268, 2352, 2744, 2800, 3168, 3360, 3402, 3520, 3528, 3750, 3872, 3920, 3969, 4352, 4752

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 A324518. 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}
   24: {1,1,1,2}
   27: {2,2,2}
   36: {1,1,2,2}
   54: {1,2,2,2}
   80: {1,1,1,1,3}
  200: {1,1,1,3,3}
  224: {1,1,1,1,1,4}
  240: {1,1,1,1,2,3}
  360: {1,1,1,2,2,3}
  405: {2,2,2,2,3}
  500: {1,1,3,3,3}
  540: {1,1,2,2,2,3}
  600: {1,1,1,2,3,3}
  625: {3,3,3,3}
  672: {1,1,1,1,1,2,4}
  675: {2,2,2,3,3}
  704: {1,1,1,1,1,1,5}
  784: {1,1,1,1,4,4}
  810: {1,2,2,2,2,3}
  900: {1,1,2,2,3,3}

Cf. A001221, A001222, A046660, A056239, A061395, A112798, A243055, A256617.

Cf. A324515, A324518, A324519, A324521, A324522, A324560, A324562.

Select[Range[2,1000],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]==Total[Last/@f]-Length[f]]&]

A061395(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

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

Original entry on oeis.org

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[#]&]!={}&]

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&]

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)).

A324560 Numbers > 1 where the minimum prime index is less than or equal to the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100

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 a certain type of integer partitions counted by A039900 (but not the type of partitions described in the name). 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:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}

Cf. A001222, A039900, A055396, A056239, A061395, A106529, A112798.

Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(min(factorset(n)))<=bigomega(n)):
select(q, [$2..100])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[2,100],PrimePi[FactorInteger[#][[1,1]]]<=PrimeOmega[#]&]

A055396(a(n)) <= A001222(a(n)).

A340600 Number of non-isomorphic balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 0, 4, 7, 16, 52, 206, 444, 1624, 5462, 19188, 62890, 215367, 765694, 2854202, 10634247, 39842786, 150669765, 581189458, 2287298588, 9157598354, 37109364812, 151970862472, 629048449881, 2635589433705, 11184718653563, 48064965080106, 208988724514022, 918639253237646, 4079974951494828

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be balanced if it has exactly as many parts as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{1,1}}  {{1,1},{1,1}}  {{1},{1},{1,1,1}}
            {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{1,2,2}}
            {{1},{2,3}}  {{1,2},{1,2}}  {{1},{1},{2,2,2}}
            {{2},{1,2}}  {{1,2},{2,2}}  {{1},{1},{2,3,3}}
                         {{1,2},{3,3}}  {{1},{1},{2,3,4}}
                         {{1,2},{3,4}}  {{1},{2},{1,2,2}}
                         {{1,3},{2,3}}  {{1},{2},{2,2,2}}
                                        {{1},{2},{2,3,3}}
                                        {{1},{2},{3,3,3}}
                                        {{1},{2},{3,4,4}}
                                        {{1},{2},{3,4,5}}
                                        {{1},{3},{2,3,3}}
                                        {{1},{4},{2,3,4}}
                                        {{2},{2},{1,2,2}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}

The version for partitions is A047993.
The co-balanced version is A319616.
The cross-balanced version is A340651.
The twice-balanced version is A340652.
The version for factorizations is A340653.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A098124 counts balanced compositions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
Cf. A001055, A006141, A007717, A316983, A320663, A324522, A340654, A340655.

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n,polcoef(G(n,n,k,y),k,y) - polcoef(G(n,n,k-1,y),k,y)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A324515 Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 40, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223

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 A324516. 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:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  23: {9}
  29: {10}
  31: {11}
  37: {12}
  40: {1,1,1,3}
  41: {13}
  43: {14}
  45: {2,2,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006141, A046660, A047993, A055396, A056239, A061395, A106529, A112798, A243055.

Cf. A324516, A324517, A324519, A324521, A324522, A324560, A324562.

filter:= proc(n) local F, Inds, t;
  if isprime(n) then return true fi;
  F:= ifactors(n)[2];
  Inds:= map(numtheory:-pi, F[..,1]);
  max(Inds) - min(Inds) = add(t[2],t=F) - nops(F)
end proc:
select(filter, [$2..300]); # Robert Israel, Nov 19 2023

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

A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

A361205 a(n) = 2*omega(n) - bigomega(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, -1, 0, 2, 1, 1, 1, 2, 2, -2, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, -1, 1, 1, 3, 1, -3, 2, 2, 2, 0, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, -1, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 2, 1, 2, 1, -4, 2, 3, 1, 1, 2, 3, 1, -1, 1, 2, 1, 1, 2, 3, 1, -1, -2, 2, 1, 2

Offset: 1

Gus Wiseman, Mar 16 2023

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

Without doubling omega we have -A046660.
Positions of 0's are A067801, counted by A239959.
Positions of negative terms are A360558, counted by A360254.
Positions of nonpositive terms are A361204, counted by A237363.
Positions of positive terms are A361393, counted by A237365.
Positions of nonnegative terms are A361395, counted by A361394.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
Cf. A061395, A067340, A111907, A324517, A324522, A326567/A326568.
Cf. A077761, A083342, A136141.

Table[2*PrimeNu[n]-PrimeOmega[n],{n,100}]

Additive with a(p^e) = 2 - e. - Amiram Eldar, Mar 26 2023

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = 2*A077761 - A083342 = A077761 - A136141 = -0.511659... . - Amiram Eldar, Oct 01 2023

cp's OEIS Frontend

A340607 Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.

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A324517 Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.

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

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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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A324560 Numbers > 1 where the minimum prime index is less than or equal to the number of prime factors counted with multiplicity.

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A340600 Number of non-isomorphic balanced multiset partitions of weight n.

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