A340596 -id:A340596 - cp's OEIS frontend

A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 2, 3, 1, 1, 2, 2, 1, 3, 1, 2, 4, 1, 1, 1, 2, 2, 2, 2, 1, 4, 2, 2, 2, 1, 1, 4, 1, 1, 4, 0, 2, 3, 1, 2, 2, 2, 1, 4, 1, 1, 4, 2, 2, 3, 1, 3, 5, 1, 1, 5, 2, 1, 2, 3, 1, 5, 2, 2, 2, 1, 2, 1, 1, 2, 4, 4, 1, 3, 1, 3, 5, 1, 1, 6

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)      (4*27)        (135)       (4*45)        (4*63)
  (5*9)     (2*6*9)       (3*45)      (12*15)       (12*21)
  (3*15)    (3*4*9)       (5*27)      (4*5*9)       (4*7*9)
  (3*3*5)   (2*2*27)      (9*15)      (2*2*45)      (6*6*7)
            (2*2*3*9)     (3*5*9)     (2*6*15)      (2*2*63)
            (2*2*3*3*3)   (3*3*15)    (3*4*15)      (2*6*21)
                          (3*3*3*5)   (2*2*5*9)     (3*4*21)
                                      (3*3*4*5)     (2*2*7*9)
                                      (2*2*3*15)    (2*3*6*7)
                                      (2*2*3*3*5)   (3*3*4*7)
                                                    (2*2*3*21)
                                                    (2*2*3*3*7)

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

Positions of 0's are A000079.
The version for partitions is A027193.
The version for prime indices is A244991.
The version looking at length instead of greatest factor is A339890.
The version that also has odd length is A340607.
The version looking at least factor is A340832.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A058695 counts partitions of odd numbers.
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.

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@*Max]],{n,100}]

A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

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

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 01 2024

nonn

All of these factorizations are co-balanced (A340596).

			The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
  (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
       (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
              (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
                      (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
                              (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
                                      (3*4*10)  (9*16)   (2*9*10)  (12*24)
                                                (12*12)  (3*4*15)  (16*18)
                                                         (3*5*12)  (2*144)
                                                         (3*6*10)

Multisets of this type are ranked by A368101, see also A368100, A355529.
For nonexistence we have A368413, complement A368414.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, see also A370592, A370593.
Subsets of this type are counted by A370638, see also A370636, A370637.
For unlabeled multiset partitions we have A370646, also A368098, A368097.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A027746 lists prime factors, A112798 indices, length A001222.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
A355741 counts ways to choose a prime factor of each prime index.
For set-systems see A367902-A367908.
Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.

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[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]

A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 0, 2, 3, 6, 20, 65, 134, 482, 1562, 4974, 15466, 51768, 179055, 631737, 2216757, 7905325, 28768472, 106852116, 402255207, 1532029660, 5902839974, 23041880550, 91129833143, 364957188701, 1478719359501, 6058859894440, 25100003070184, 105123020009481, 445036528737301

Offset: 0

Gus Wiseman, Feb 07 2021

nonn

We define a multiset partition to be twice-balanced if all of the following are equal:
(1) the number of parts;
(2) the number of distinct vertices;
(3) the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 6 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{2,3,3}}
            {{2},{1,2}}  {{1,2},{1,2}}  {{1},{2},{2,3,3}}
                         {{1,2},{2,2}}  {{1},{2},{3,3,3}}
                                        {{1},{3},{2,3,3}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}

The co-balanced version is A319616.
The singly balanced version is A340600.
The cross-balanced version is A340651.
The version for factorizations is A340655.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A303975 counts distinct prime factors in prime indices.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340653 counts balanced factorizations.
- A340657/A340656 list numbers with/without a twice-balanced factorization.
Cf. A001055, A001221, A001222, A007717, A316983, A320663, A339888, A340597, A340599, A340654.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(m,n,k,y=1)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, y^t*subst(x*Polrev(K(q, t, min(k,n\t))), x, x^t)/t, O(x*x^n)))); s/m!}
seq(n)={Vec(1 + sum(k=1,n, polcoef(G(k,n,k,y) - G(k-1,n,k,y) - G(k,n,k-1,y) + G(k-1,n,k-1,y), k, y)))} \\ Andrew Howroyd, Jan 15 2024

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

A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose number of factors is divisible by their least common multiple.

			The a(n) factorizations for n = 8192, 46656, 73728:
  2*2*2*2*2*4*8*8          6*6*6*6*6*6              2*2*2*2*2*2*2*2*2*4*6*6
  2*2*2*2*4*4*4*8          2*2*2*2*2*2*3*3*3*3*3*3  2*2*2*2*2*2*2*2*3*4*4*6
  2*2*2*4*4*4*4*4                                   2*2*2*2*2*2*2*3*3*4*4*4
  2*2*2*2*2*2*2*2*2*2*2*4                           2*2*2*2*2*2*2*2*2*2*6*12
                                                    2*2*2*2*2*2*2*2*2*3*4*12

The version for partitions is A340693, with reciprocal version A143773.
Positions of nonzero terms are A340852.
The reciprocal version is A340853.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even numbers, even-length case A340786.
A340831/A340832 count factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.

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],And@@IntegerQ/@(Length[#]/#)&]],{n,100}]

A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose greatest common divisor is a multiple of the number of factors.

			The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
  2   4     16    48     96     144     216      240     432
      2*2   2*8   6*8    2*48   2*72    4*54     4*60    6*72
            4*4   2*24   4*24   4*36    6*36     6*40    8*54
                  4*12   6*16   6*24    12*18    8*30    12*36
                         8*12   8*18    2*108    10*24   18*24
                                12*12   6*6*6    12*20   2*216
                                        3*3*24   2*120   4*108
                                        3*6*12           3*3*48
                                                         3*6*24
                                                         6*6*12
                                                         3*12*12

Positions of 1's are A048103.
Positions of terms > 1 are A100716.
The version for partitions is A143773 (A316428).
The reciprocal for partitions is A340693 (A340606).
The version for strict partitions is A340830.
The reciprocal version is A340851.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even factors, even-length case A340786.
A340831/A340832 counts factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.

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],n>1&&Divisible[GCD@@#,Length[#]]&]],{n,100}]

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

Gus Wiseman, Jan 28 2021

nonn

			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

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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.

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@@#&]!={}&]

More terms from Chai Wah Wu, Feb 01 2021

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 11, 26, 77, 220, 677, 2098, 6756, 22101, 74264, 253684, 883795, 3130432, 11275246, 41240180, 153117873, 576634463, 2201600769, 8517634249, 33378499157, 132438117118, 531873247805, 2161293783123, 8883906870289, 36928576428885, 155196725172548, 659272353608609, 2830200765183775

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be cross-balanced if it uses exactly as many distinct vertices as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,2}}    {{1,2},{1,2}}
                    {{1},{1},{1}}  {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The co-balanced version is A319616.
The balanced version is A340600.
The twice-balanced version is A340652.
The version for factorizations is A340654.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A001055, A007717, A064174, A316983, A320663, A324522, A339888, A340655.

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

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

A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

Original entry on oeis.org

2, 8, 12, 16, 32, 48, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 224, 240, 256, 280, 288, 320, 336, 384, 392, 432, 448, 480, 512, 576, 640, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1008, 1024, 1056, 1120, 1152, 1200, 1248, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

			The initial terms and a valid factorization of each:
      2 = 2           168 = 3*7*8        512 = 2*2*2*2*32
      8 = 2*4         192 = 2*2*3*16     576 = 2*2*9*16
     12 = 3*4         200 = 5*5*8        640 = 2*2*10*16
     16 = 4*4         224 = 4*7*8        672 = 2*3*7*16
     32 = 2*2*8       240 = 5*6*8        704 = 2*2*11*16
     48 = 2*3*8       256 = 2*2*4*16     720 = 3*3*5*16
     64 = 2*4*8       280 = 5*7*8        768 = 2*2*2*3*32
     72 = 3*3*8       288 = 2*3*3*16     800 = 2*5*5*16
     80 = 2*5*8       320 = 2*2*5*16     832 = 2*2*13*16
     96 = 2*6*8       336 = 6*7*8        864 = 2*3*9*16
    112 = 2*7*8       384 = 2*2*6*16     896 = 2*2*14*16
    120 = 3*5*8       392 = 7*7*8        960 = 2*2*15*16
    128 = 2*2*2*16    432 = 3*3*3*16    1008 = 3*3*7*16
    144 = 3*6*8       448 = 2*2*7*16    1024 = 2*2*2*4*32
    160 = 4*5*8       480 = 2*3*5*16    1056 = 2*3*11*16

Partitions of the prescribed type are counted by A340611.
The conjugate version is A340689.
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.

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[#],2^Length[#]==Max@@#&]!={}&]

A370815 Number of integer factorizations of n into unordered factors > 1, such that only one set can be obtained by choosing a different divisor of each factor.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 06 2024

nonn

			The a(432) = 3 factorizations: (2*2*3*4*9), (2*3*3*4*6), (2*6*6*6).

For partitions and prime factors we have A370594, ranks A370647.
Partitions of this type are counted by A370595, ranks A370810.
For prime factors we have A370645, subsets A370584.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A239312 counts condensed partitions, ranks A355740, complement A370320.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A368414 counts factor-choosable factorizations, complement A368413.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A340596, A340653, A355529, A355739, A368110, A370592, A370638, A370803.

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[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,100}]

cp's OEIS Frontend

A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

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A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

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A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

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A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

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A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

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

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A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

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A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

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A370815 Number of integer factorizations of n into unordered factors > 1, such that only one set can be obtained by choosing a different divisor of each factor.

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