A340387 -id:A340387 - cp's OEIS frontend

A347045 Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 3, 2, 1, 4, 5, 2, 1, 1, 1, 1, 1, 1, 3, 2, 5, 4, 1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 3, 1, 1, 6, 5, 4, 3, 2, 1, 4, 1, 2, 1, 8, 5, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 1, 9, 2, 1, 4, 5, 2, 3

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

			The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 6.

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

The smallest divisor without the condition is A020639 (greatest: A006530).
Positions of 1's are A026424.
Positions of even terms are A063745 = 2*A026424.
The case of powers of 2 is A072345.
Positions of 2's are A100484.
Divisors of this type are counted by A345957 (rounded: A096825).
The rounded version is A347043.
The greatest divisor of this type is A347046 (rounded: A347044).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A001747, A027746, A038548, A056924, A063962, A106529, A140271, A244991, A324522, A335433, A335448.

Table[If[#=={},1,Min[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[1 ;; np/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[1:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347045(n):
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 1 if r else prod(fs[:q]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=1..A001222(n)/2} A027746(n,k) if A001222(n) is even, and 1 otherwise. - Amiram Eldar, Nov 02 2024

A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 3, 5, 1, 1, 1, 7, 5, 4, 1, 1, 1, 1, 7, 11, 1, 6, 5, 13, 1, 1, 1, 1, 1, 1, 11, 17, 7, 9, 1, 19, 13, 10, 1, 1, 1, 1, 1, 23, 1, 1, 7, 1, 17, 1, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 1, 8, 13, 1, 1, 1, 23, 1, 1, 1, 1, 37, 1, 1, 11, 1, 1, 1, 9

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

Problem: What are the positions of last appearances > 1?

			The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 15.

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

The greatest divisor without the condition is A006530 (smallest: A020639).
Positions of 1's are A026424.
The case of powers of 2 is A072345.
Positions of first appearances are A123667 (sorted: A123666).
Divisors of this type are counted by A345957 (rounded: A096825).
The rounded version is A347044.
The smallest divisor of this is A347045 (rounded: A347043).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A038548, A060775, A106529, A244991, A335433, A335448.

Table[If[#=={},1,Max[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[np/2+1 ;; np]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[-1:0:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347046(n):
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 1 if r else prod(fs[q:]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=A001222(n)/2+1..A001222(n)} A027746(n,k) if A001222(n) is even, and 1 otherwise. - Amiram Eldar, Nov 02 2024

A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 6, 6, 9, 4, 3, 7, 2, 6, 10, 6, 11, 1, 7, 8, 7, 6, 12, 9, 8, 5, 13, 7, 14, 7, 7, 10, 15, 4, 4, 7, 9, 8, 16, 5, 8, 6, 10, 11, 17, 7, 18, 12, 8, 1, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 5

Offset: 1

Gus Wiseman, Jul 06 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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 prime indices of 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.

Positions of first appearances are 1 followed by the primes A000040.
Positions of 1 are A000079 (powers of two) except 1.
Positions of 2 are A000244 (powers of three) except 1.
Positions of 3 are {6} U A000351 (six or powers of five) except 1.
For number of runs instead of sum of run-compression we have A373957.
For prime factors instead of indices we have A374250.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A001223, A001414, A037201, A116608, A124767, A316413, A333755, A340387, A344415, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) iff n belongs to A335433 (the separable case), complement A335448.

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

Gus Wiseman, Jan 28 2021

nonn

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

			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)

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.

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

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

A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 22, 25, 27, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138

Offset: 1

Gus Wiseman, May 19 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.

Also Heinz numbers of integer partitions of even numbers m with at most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
      1: {}          37: {12}        75: {2,3,3}
      3: {2}         39: {2,6}       76: {1,1,8}
      7: {4}         43: {14}        79: {22}
      9: {2,2}       46: {1,9}       81: {2,2,2,2}
     10: {1,3}       49: {4,4}       82: {1,13}
     13: {6}         52: {1,1,6}     84: {1,1,2,4}
     19: {8}         53: {16}        85: {3,7}
     21: {2,4}       55: {3,5}       87: {2,10}
     22: {1,5}       57: {2,8}       88: {1,1,1,5}
     25: {3,3}       61: {18}        89: {24}
     27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
     28: {1,1,4}     63: {2,2,4}     91: {4,6}
     29: {10}        66: {1,2,5}     94: {1,15}
     30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
     34: {1,7}       71: {20}       101: {26}
For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.

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

These are the Heinz numbers of partitions counted by A209816 and A110618.
A subset of A300061 (sum of prime indices is even).
The conjugate version appears to be A320924 (allowing odd weights: A322109).
The case of equality is A340387.
Allowing odd weights gives A344291.
The 5-smooth case is A344295, or A344293 allowing odd weights.
The opposite version allowing odd weights is A344296.
The conjugate opposite version allowing odd weights is A344414.
The case of equality in the conjugate case is A344415.
The conjugate opposite version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

filter:= proc(n) local F,a,t;
  F:= ifactors(n)[2];
  a:= add((numtheory:-pi(t[1])-2)*t[2],t=F);
  a::even and a >= 0
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 10 2024

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&]

Members m of A300061 such that A056239(m) >= 2*A001222(m).

A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Oct 10 2023

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 integer partitions containing n/2, where n is the sum of all parts.

			The prime indices of 90 are {1,2,2,3}, with sum 8 and twice maximum 6, so 90 is in the sequence.

Partitions of this type are counted by A086543.
For length instead of maximum we have the complement of A340387.
The complement is A344415, counted by A035363.
A001221 counts distinct prime factors, A001222 with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A001414, A100959, A238628, A300061, A301987, A316413, A320924, A344414, A344416, A366318.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[prix[#]]!=Total[prix[#]]/2&]

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@@#&]!={}&]

A343938 Twice the number of prime factors of n minus the sum of prime indices of n, both counted with multiplicity.

Original entry on oeis.org

0, 1, 0, 2, -1, 1, -2, 3, 0, 0, -3, 2, -4, -1, -1, 4, -5, 1, -6, 1, -2, -2, -7, 3, -2, -3, 0, 0, -8, 0, -9, 5, -3, -4, -3, 2, -10, -5, -4, 2, -11, -1, -12, -1, -1, -6, -13, 4, -4, -1, -5, -2, -14, 1, -4, 1, -6, -7, -15, 1, -16, -8, -2, 6, -5, -2, -17, -3, -7

Offset: 1

Gus Wiseman, Aug 04 2021

sign

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.

			For n = 1050 we have 5 prime indices {1,2,3,3,4}, so a(1050) = 10 - 13 = -3.

First appearances are the elements of A174090 except for 3.
Positions of zeros are A340387.
Positions of nonpositive terms are A344291.
Positions of nonnegative terms are A344296.
A001222 counts prime indices.
A056239 adds up prime indices.
A112798 lists prime indices.
Cf. A001221, A003963, A061775, A275024, A279789, A299757, A302242.

Table[2*PrimeOmega[n]-Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]],{n,100}]

Totally additive with a(prime(k)) = 2 - k.

A344295 Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.

Original entry on oeis.org

1, 3, 9, 10, 25, 27, 30, 75, 81, 90, 100, 225, 243, 250, 270, 300, 625, 675, 729, 750, 810, 900, 1000, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 5625, 6075, 6250, 6561, 6750, 7290, 7500, 8100, 9000, 10000, 15625, 16875, 18225, 18750, 19683, 20250, 21870

Offset: 1

Gus Wiseman, May 15 2021

nonn

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 sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     25: {3,3}
     27: {2,2,2}
     30: {1,2,3}
     75: {2,3,3}
     81: {2,2,2,2}
     90: {1,2,2,3}
    100: {1,1,3,3}
    225: {2,2,3,3}
    243: {2,2,2,2,2}
    250: {1,3,3,3}
    270: {1,2,2,2,3}
    300: {1,1,2,3,3}

These partitions are counted by A001399.
Allowing any number of parts and sum gives A051037.
Allowing parts > 3 and any length gives A300061.
Not requiring the sum of prime indices to be even gives A344293.
Allowing any number of parts (but still with even sum) gives A344297.
Allowing parts > 3 gives A344413.
A001358 lists semiprimes.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n of length n/2, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A110618 counts partitions of n with at most n/2 parts, ranked by A344291.
A344414 counts partitions of n with all parts >= n/2, ranked by A344296.
Cf. A030229, A080193, A244990, A261144, A266755, A279622, A334433, A344294.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&&Max@@primeMS[#]<=3&]

Intersection of A300061 (even Heinz weight), A344291 (Omega > half Heinz weight), and A051037 (5-smooth).

cp's OEIS Frontend

A347045 Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

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A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

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A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

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

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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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A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

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A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

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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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