A340609 -id:A340609 - cp's OEIS frontend

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

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

Gus Wiseman, Feb 01 2021

nonn

			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

FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.

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).

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

A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 16, 20, 24, 32, 36, 50, 54, 56, 64, 81, 84, 96, 125, 126, 128, 144, 160, 176, 189, 196, 216, 240, 256, 294, 324, 360, 384, 400, 416, 441, 486, 512, 540, 576, 600, 624, 686, 729, 810, 864, 896, 900, 936, 968, 1000, 1024, 1029, 1040, 1088, 1215

Offset: 1

Gus Wiseman, Jan 24 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:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  50: {1,3,3}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  81: {2,2,2,2}
  84: {1,1,2,4}
  96: {1,1,1,1,1,2}

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428).
These partitions are counted by A340693.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A003963 multiplies together the prime indices of n.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length divides n (A316413).
A056239 adds up the prime indices of n.
A061395 selects the maximum prime index.
A067538 counts partitions of n whose maximum divides n (A326836).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 = partitions whose length is divisible by their maximum (A340609).
A168659 = partitions whose maximum is divisible by their length (A340610).
A289509 lists numbers with relatively prime prime indices.
A326842 = partitions of n whose length and parts all divide n (A326847).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340852 have a factorization with factors dividing length.
Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]

A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 1, 3, 5, 2, 6, 2, 3, 1, 7, 2, 8, 1, 2, 5, 9, 1, 3, 3, 2, 4, 10, 1, 11, 1, 5, 7, 2, 1, 12, 4, 3, 3, 13, 4, 14, 5, 1, 9, 15, 2, 2, 1, 7, 2, 16, 1, 5, 1, 4, 5, 17, 3, 18, 11, 4, 1, 3, 5, 19, 7, 9, 4, 20, 2, 21, 6, 1, 8, 5, 2, 22, 3, 1, 13, 23, 1, 7, 7, 5, 5, 24, 3, 3, 3, 11, 15, 4, 1, 25, 4, 5, 3

Offset: 1

Gus Wiseman, Feb 19 2023

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 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 3.

Positions of 1's are A340609, a superset of A106529.
For twice median instead of mean we have A360555.
The denominator is A360615.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
A124010 gives prime signature, mean A088529/A088530.
A316413 lists numbers with integer mean prime index, complement A348551.
A326567/A326568 gives mean of prime indices.
Cf. A334201, A340610, A360008, A360556, A360557, A360688, A366785.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Numerator[Mean[Differences[Prepend[prix[n],0]]]]],{n,100}]

A360614(n) = if(1==n,0, my(u=primepi(vecmax(factor(n)[, 1]))); (u/gcd(u, bigomega(n)))); \\ Antti Karttunen, Oct 23 2023

Numerator of A061395(n)/A001222(n).

a(1) = 0; and for n >= 1, a(n) = A061395(n) / A366785(n) = A061395(n) / gcd(A001222(n), A061395(n)). - Antti Karttunen, Oct 23 2023

Data section extended up to a(100) by Antti Karttunen, Oct 23 2023

A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 19 2023

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 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 4.

Winston de Greef, Table of n, a(n) for n = 1..10000

Positions of 1's are A340610
The numerator is A360614.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
A124010 gives prime signature, mean A088529/A088530.
A316413 lists numbers with integer mean prime index, complement A348551.
A326567/A326568 gives mean of prime indices.
Cf. A106529, A334201, A340609, A360008, A360555, A360556, A360558.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Denominator[Mean[Differences[Prepend[prix[n],0]]]]],{n,100}]

a(n) = if (n==1, 0, my(f=factor(n)); denominator(primepi(vecmax(f[, 1]))/ bigomega(f))); \\ Michel Marcus, Feb 20 2023

Denominator of A061395(n)/A001222(n), for n>1.

A347045 Smallest 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, 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

A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892

Offset: 0

Gus Wiseman, Jan 23 2021

nonn

The only strict partitions counted are (), (1), and (2,1).

Is there a simple generating function?

			The a(1) = 1 through a(9) = 7 partitions:
  1  11  21   22    311    2211    331      2222      333
         111  1111  2111   111111  2221     4211      4221
                    11111          4111     221111    51111
                                   211111   311111    222111
                                   1111111  11111111  321111
                                                      21111111
                                                      111111111

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428), with strict case A340830.
The case where length also divides n is A326842 (A326847).
The Heinz numbers of these partitions are A340606.
The version for factorizations is A340851, with reciprocal version A340853.
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).
A067539 counts partitions with integer geometric mean (A326623).
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).
A330950 = partitions of n whose Heinz number is divisible by n (A324851).
Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]

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

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}]

A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

Also Heinz numbers of strict integer partitions whose greatest part is divisible by their number of parts. These partitions are counted by A340828.

			The sequence of terms together with their prime indices begins:
      2: {1}         31: {11}       71: {20}
      3: {2}         35: {3,4}      73: {21}
      5: {3}         37: {12}       74: {1,12}
      6: {1,2}       38: {1,8}      78: {1,2,6}
      7: {4}         39: {2,6}      79: {22}
     11: {5}         41: {13}       83: {23}
     13: {6}         43: {14}       86: {1,14}
     14: {1,4}       47: {15}       87: {2,10}
     17: {7}         53: {16}       89: {24}
     19: {8}         57: {2,8}      91: {4,6}
     21: {2,4}       58: {1,10}     95: {3,8}
     23: {9}         59: {17}       97: {25}
     26: {1,6}       61: {18}      101: {26}
     29: {10}        65: {3,6}     103: {27}
     30: {1,2,3}     67: {19}      106: {1,16}

Note: Heinz number sequences are given in parentheses below.
The case of equality, and the reciprocal version, are both A002110.
The non-strict reciprocal version is A168659 (A340609).
The non-strict version is A168659 (A340610).
These are the Heinz numbers of partitions counted by A340828.
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up the prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413/A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A257541 gives the rank of the partition with Heinz number n.
A340830 counts strict partitions whose parts are multiples of the length.
Cf. A039900, A047993 (A106529), A064174, A143773 (A316428), A326837, A326849 (A326848), A340599, A340691.

Select[Range[2,100],SquareFreeQ[#]&&Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A340691 Greatest image of A001222 over the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

For the initial term, we assume the empty set has maximum image 0.

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 4070 are {1,3,5,12} -> {0,1,1,3}, so a(4070) = 3.
The prime indices of 8892 are {1,1,2,2,6,8} -> {0,0,1,1,2,3} so a(8892) = 3.

Positions of first appearances are A033844.
Positions of 0's are A000079.
Positions of terms <= 1 are A302540.
Positions of 1's are A302540 \ A000079.
The version for minimum is A340928.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices.
A061395 selects the greatest prime index.
A072233 counts partitions by sum and maximum.
A112798 lists the prime indices of each positive integer.
A303975 counts distinct prime factors in the product of prime indices.
Cf. A001222, A006530, A062447, A106529, A244990, A244991, A324522, A340606, A340609, A340610, A340856.

Table[If[n==1,0,Max@@PrimeOmega/@PrimePi/@First/@FactorInteger[n]],{n,100}]

cp's OEIS Frontend

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

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A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

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A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

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A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

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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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A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

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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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