A340601 -id:A340601 - cp's OEIS frontend

A340785 Number of factorizations of 2n into even factors > 1.

1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 7, 1, 3, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 3, 1, 11, 1, 2, 1, 6, 1, 2, 1, 7, 1, 3, 1, 4, 1, 2, 1, 12, 1, 3, 1, 4, 1, 3, 1, 7, 1, 2, 1, 7, 1, 2, 1, 15, 1, 3, 1, 4, 1, 3, 1, 12, 1, 2, 1, 4, 1, 3, 1, 12, 1, 2, 1, 7, 1

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

			The a(n) factorizations for n = 2*2, 2*4, 2*8, 2*12, 2*16, 2*32, 2*36, 2*48 are:
  4    8      16       24     32         64           72      96
  2*2  2*4    2*8      4*6    4*8        8*8          2*36    2*48
       2*2*2  4*4      2*12   2*16       2*32         4*18    4*24
              2*2*4    2*2*6  2*2*8      4*16         6*12    6*16
              2*2*2*2         2*4*4      2*4*8        2*6*6   8*12
                              2*2*2*4    4*4*4        2*2*18  2*6*8
                              2*2*2*2*2  2*2*16               4*4*6
                                         2*2*2*8              2*2*24
                                         2*2*4*4              2*4*12
                                         2*2*2*2*4            2*2*4*6
                                         2*2*2*2*2*2          2*2*2*12
                                                              2*2*2*2*6

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

Note: A-numbers of Heinz-number sequences are in parentheses below.
The version for partitions is A035363 (A066207).
The odd version is A340101.
The even length case is A340786.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
A316439 counts factorizations by product and length
A340102 counts odd-length factorizations of odd numbers into odd factors.
- Even -
A027187 counts partitions of even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum.
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A050320, A066208, A160786, A174725, A320655, A320656, A339890, A340851.
Even bisection of A349906.

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],Select[#,OddQ]=={}&]],{n,2,100,2}]

A349906(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A349906(n/d, d))); (s));
A340785(n) = A349906(2*n); \\ Antti Karttunen, Dec 13 2021

a(n) = A349906(2*n). - Antti Karttunen, Dec 13 2021

A340788 Heinz numbers of integer partitions of negative rank.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405

Offset: 1

Gus Wiseman, Jan 29 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 Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
      4: (1,1)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)

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.
These partitions are counted by A064173.
The odd case is A101707 is (A340929).
The even case is A101708 is (A340930).
The positive version is (A340787).
A001222 counts prime factors.
A061395 selects the maximum prime index.
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.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]

For all terms A061395(a(n)) < A001222(a(n)).

A340605 Heinz numbers of integer partitions of even positive rank.

Original entry on oeis.org

5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 with their Heinz numbers begins:
      5: (3)         57: (8,2)       97: (25)
     11: (5)         58: (10,1)      99: (5,2,2)
     14: (4,1)       59: (17)       102: (7,2,1)
     17: (7)         65: (6,3)      103: (27)
     21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
     23: (9)         67: (19)       106: (16,1)
     26: (6,1)       68: (7,1,1)    109: (29)
     31: (11)        73: (21)       110: (5,3,1)
     35: (4,3)       74: (12,1)     111: (12,2)
     38: (8,1)       83: (23)       122: (18,1)
     39: (6,2)       86: (14,1)     124: (11,1,1)
     41: (13)        87: (10,2)     127: (31)
     44: (5,1,1)     91: (6,4)      129: (14,2)
     47: (15)        92: (9,1,1)    133: (8,4)
     49: (4,4)       95: (8,3)      137: (33)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
Allowing any positive rank gives A064173 (A340787).
The odd version is counted by A101707 (A340604).
These partitions are counted by A101708.
The not necessarily positive case is counted by A340601 (A340602).
A001222 counts prime indices.
A061395 gives maximum prime index.
A072233 counts partitions by sum and length.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340692 counts partitions of odd rank (A340603).
- Even -
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
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.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],EvenQ[rk[#]]&&rk[#]>0&]

A061395(a(n)) - A001222(a(n)) is even and positive.

A340604 \/ A340605 = A340787.

A340603 Heinz numbers of integer partitions of odd rank.

Original entry on oeis.org

3, 4, 7, 10, 12, 13, 15, 16, 18, 19, 22, 25, 27, 28, 29, 33, 34, 37, 40, 42, 43, 46, 48, 51, 52, 53, 55, 60, 61, 62, 63, 64, 69, 70, 71, 72, 76, 77, 78, 79, 82, 85, 88, 89, 90, 93, 94, 98, 100, 101, 105, 107, 108, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 with their Heinz numbers begins:
      3: (2)           33: (5,2)           63: (4,2,2)
      4: (1,1)         34: (7,1)           64: (1,1,1,1,1,1)
      7: (4)           37: (12)            69: (9,2)
     10: (3,1)         40: (3,1,1,1)       70: (4,3,1)
     12: (2,1,1)       42: (4,2,1)         71: (20)
     13: (6)           43: (14)            72: (2,2,1,1,1)
     15: (3,2)         46: (9,1)           76: (8,1,1)
     16: (1,1,1,1)     48: (2,1,1,1,1)     77: (5,4)
     18: (2,2,1)       51: (7,2)           78: (6,2,1)
     19: (8)           52: (6,1,1)         79: (22)
     22: (5,1)         53: (16)            82: (13,1)
     25: (3,3)         55: (5,3)           85: (7,3)
     27: (2,2,2)       60: (3,2,1,1)       88: (5,1,1,1)
     28: (4,1,1)       61: (18)            89: (24)
     29: (10)          62: (11,1)          90: (3,2,2,1)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A340692.
The complement is A340602, counted by A340601.
The case of positive rank is A340604.
- Rank -
A001222 gives number of prime indices.
A047993 counts partitions of rank 0 (A106529).
A061395 gives maximum prime index.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
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.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340608, A340609, A340610.

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

A061395(a(n)) - A001222(a(n)) is odd.

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Jan 29 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 Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

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.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
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).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

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

For all terms A061395(a(n)) > A001222(a(n)).

A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 3, 4, 4, 4, 8, 11, 11, 11, 20, 27, 29, 31, 48, 65, 70, 74, 109, 145, 160, 172, 238, 314, 345, 372, 500, 649, 721, 782, 1019, 1307, 1451, 1577, 2015, 2552, 2841, 3098, 3885, 4867, 5418, 5914, 7318, 9071, 10109, 11050

Offset: 0

Gus Wiseman, Jun 26 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. By conjugation, this is also (-1)^(r-1) times the number of odd parts, where r is the greatest part. So a(n) is the number of integer partitions of n of even rank with the same number of odd parts as their conjugate.

			The a(5) = 1 through a(12) = 11 partitions:
  (311)  (321)  (43)    (44)    (333)    (541)    (65)      (66)
                (2221)  (332)   (531)    (4321)   (4322)    (552)
                (4111)  (2222)  (32211)  (32221)  (4331)    (4332)
                        (4211)  (51111)  (52111)  (4421)    (4422)
                                                  (6311)    (4431)
                                                  (222221)  (6411)
                                                  (422111)  (33222)
                                                  (611111)  (53211)
                                                            (222222)
                                                            (422211)
                                                            (621111)

The non-reverse version is A277103.
Comparing even parts to odd conjugate parts gives A277579.
Comparing signs only gives A340601.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A006330, A027187, A027193, A236559, A239829, A257991, A344607, A344608, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],sats[#]==sats[conj[#]]&]],{n,0,15}]

A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

Original entry on oeis.org

3, 7, 9, 13, 15, 19, 21, 27, 29, 33, 37, 39, 43, 45, 49, 51, 53, 57, 61, 63, 69, 71, 75, 77, 79, 81, 87, 89, 91, 93, 99, 101, 105, 107, 111, 113, 117, 119, 123, 129, 131, 133, 135, 139, 141, 147, 151, 153, 159, 161, 163, 165, 169, 171, 173, 177, 181, 183

Offset: 1

Gus Wiseman, Feb 12 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. 1 has no prime indices so is not counted.

			The sequence of terms together with their prime indices begins:
      3: {2}         51: {2,7}         99: {2,2,5}
      7: {4}         53: {16}         101: {26}
      9: {2,2}       57: {2,8}        105: {2,3,4}
     13: {6}         61: {18}         107: {28}
     15: {2,3}       63: {2,2,4}      111: {2,12}
     19: {8}         69: {2,9}        113: {30}
     21: {2,4}       71: {20}         117: {2,2,6}
     27: {2,2,2}     75: {2,3,3}      119: {4,7}
     29: {10}        77: {4,5}        123: {2,13}
     33: {2,5}       79: {22}         129: {2,14}
     37: {12}        81: {2,2,2,2}    131: {32}
     39: {2,6}       87: {2,10}       133: {4,8}
     43: {14}        89: {24}         135: {2,2,2,3}
     45: {2,2,3}     91: {4,6}        139: {34}
     49: {4,4}       93: {2,11}       141: {2,15}

These partitions are counted by A026805.
Looking at length or at maximum gives A028260/A244990, counted by A027187.
If all prime indices are even we get A066207, counted by A035363.
The complement is {1} \/ A340932, counted by A026804.
A001222 counts prime factors.
A005843 lists even numbers.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058695 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A006141, A030229, A067661, A101708, A236913,  A339846, A340601, A340602, A340605, A340854/A340855.

Select[Range[2,100],EvenQ[PrimePi[FactorInteger[#][[1,1]]]]&]

A055396(a(n)) belongs to A005843.

Closed under multiplication.

A340786 Number of factorizations of 4n into an even number of even factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Jan 31 2021

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
  4*6   6*8      2*48      2*72      4*60      4*80          40*50
  2*12  2*24     4*24      4*36      6*40      8*40          4*500
        4*12     6*16      6*24      8*30      10*32         8*250
        2*2*2*6  8*12      8*18      10*24     16*20         10*200
                 2*2*4*6   12*12     12*20     2*160         20*100
                 2*2*2*12  2*2*6*6   2*120     2*2*2*40      2*1000
                           2*2*2*18  2*2*2*30  2*2*4*20      2*2*10*50
                                     2*2*6*10  2*2*8*10      2*2*2*250
                                               2*4*4*10      2*10*10*10
                                               2*2*2*2*2*10

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

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of ones are 1 and A000040, or A008578.
A version for partitions is A027187 (A028260).
Allowing odd length gives A108501 (bisection of A340785).
Allowing odd factors gives A339846.
An odd version is A340102.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
- Even -
A027187 counts partitions of even maximum (A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

g:= proc(n, m, p)
 option remember;
 local F,r,x,i;
 # number of factorizations of n into even factors > m with number of factors == p (mod 2)
 if n = 1 then if p = 0 then return 1 else return 0 fi fi;
 if m > n  or n::odd then return 0 fi;
 F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)),list));
 r:= 0;
 for x in F do
   for i from 1 while n mod x^i = 0 do
     r:= r + procname(n/x^i, x, (p+i) mod 2)
 od od;
 r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n],EvenQ[Length[#]]&&Select[#,OddQ]=={}&]],{n,100}]

A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022

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}]

A363233 Number of partitions of n with rank a multiple of 4.

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 5, 4, 10, 8, 16, 17, 29, 29, 48, 53, 81, 89, 130, 149, 208, 238, 325, 381, 506, 592, 770, 910, 1165, 1374, 1738, 2057, 2571, 3038, 3761, 4451, 5461, 6447, 7855, 9270, 11219, 13214, 15899, 18703, 22386, 26276, 31306, 36691, 43525, 50902, 60149, 70221, 82679, 96325

Offset: 1

Seiichi Manyama, May 23 2023

nonn

Cf. A000041, A328988, A340601, A363237, A363238, A363239.

b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 4)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..54);  # Alois P. Heinz, May 23 2023

my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(4*k))/(1-x^(4*k))))

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(4*k)) / (1-x^(4*k)).

cp's OEIS Frontend

A340785 Number of factorizations of 2n into even factors > 1.

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A340788 Heinz numbers of integer partitions of negative rank.

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A340605 Heinz numbers of integer partitions of even positive rank.

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A340603 Heinz numbers of integer partitions of odd rank.

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A340787 Heinz numbers of integer partitions of positive rank.

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A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

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A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

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A340786 Number of factorizations of 4n into an even number of even factors > 1.

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