A300272 -id:A300272 - cp's OEIS frontend

A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.

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

Offset: 0

Gus Wiseman, Dec 30 2020

nonn

			The factorizations for 2n + 1 = 135, 225, 315, 405, 675, 1155, 1215:
  135      225      315      405         675         1155      1215
  3*5*9    5*5*9    5*7*9    5*9*9       3*3*75      3*5*77    3*5*81
  3*3*15   3*3*25   3*3*35   3*3*45      3*5*45      3*7*55    3*9*45
           3*5*15   3*5*21   3*5*27      3*9*25      5*7*33    5*9*27
                    3*7*15   3*9*15      5*5*27      3*11*35   9*9*15
                             3*3*3*3*5   5*9*15      5*11*21   3*15*27
                                         3*15*15     7*11*15   3*3*135
                                         3*3*3*5*5             3*3*3*5*9
                                                               3*3*3*3*15

The version for partitions is A160786, ranked by A300272.
The not necessarily odd-length version is A340101.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A316439 counts factorizations by product and length.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=0, 0, g(2*n+1$2, 1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ[Length[#]]&&OddQ[Times@@#]&]],{n,1,100,2}];

A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

Original entry on oeis.org

2, 5, 6, 11, 14, 17, 18, 23, 26, 31, 35, 38, 41, 42, 47, 54, 58, 59, 65, 67, 73, 74, 78, 83, 86, 95, 97, 98, 103, 106, 109, 114, 122, 126, 127, 137, 142, 143, 145, 149, 157, 158, 162, 167, 174, 178, 179, 182, 185, 191, 197, 202, 209, 211, 214, 215, 222, 226

Offset: 1

Gus Wiseman, Feb 12 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only odd prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)         54: (2,2,2,1)    109: (29)
      5: (3)         58: (10,1)       114: (8,2,1)
      6: (2,1)       59: (17)         122: (18,1)
     11: (5)         65: (6,3)        126: (4,2,2,1)
     14: (4,1)       67: (19)         127: (31)
     17: (7)         73: (21)         137: (33)
     18: (2,2,1)     74: (12,1)       142: (20,1)
     23: (9)         78: (6,2,1)      143: (6,5)
     26: (6,1)       83: (23)         145: (10,3)
     31: (11)        86: (14,1)       149: (35)
     35: (4,3)       95: (8,3)        157: (37)
     38: (8,1)       97: (25)         158: (22,1)
     41: (13)        98: (4,4,1)      162: (2,2,2,2,1)
     42: (4,2,1)    103: (27)         167: (39)
     47: (15)       106: (16,1)       174: (10,2,1)

These partitions are counted by A035363 (shifted left once).
Terms of A340932 can be factored into elements of this sequence.
The even version is A341447.
A001222 counts prime factors.
A005408 lists odd numbers.
A026804 counts partitions whose smallest part is odd.
A027193 counts odd-length partitions, ranked by A026424.
A031368 lists odd-indexed primes.
A032742 selects largest proper divisor.
A055396 selects smallest prime index.
A056239 adds up prime indices.
A058695 counts partitions of odd numbers, ranked by A300063.
A061395 selects largest prime index.
A066207 lists numbers with all even prime indices.
A066208 lists numbers with all odd prime indices.
A112798 lists the prime indices of each positive integer.
A244991 lists numbers whose greatest prime index is odd.
A340932 lists numbers whose smallest prime index is odd.
Cf. A006141, A160786, A257991, A257992, A300272, A340604, A340607, A340854/A340855, A340933.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[First[primeMS[#]]]&&And@@EvenQ[Rest[primeMS[#]]]&]

Also numbers n > 1 such that A055396(n) is odd and A032742(n) belongs to A066207.

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 25 2021

nonn

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

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

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

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

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

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

A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 6, 3, 10, 7, 18, 15, 30, 28, 51, 50, 82, 87, 134, 145, 211, 235, 331, 375, 510, 586, 779, 901, 1172, 1366, 1750, 2045, 2581, 3026, 3778, 4433, 5476, 6430, 7878, 9246, 11240, 13189, 15931, 18670, 22417, 26242, 31349, 36646, 43567, 50854

Offset: 1

Gus Wiseman, Jan 08 2021

nonn

			The a(3) = 2 through a(10) = 7 partitions:
  3     5       321   7         332     9           532
  111   311           322       521     333         541
        11111         331       32111   522         721
                      511               531         32221
                      31111             711         33211
                      1111111           32211       52111
                                        33111       3211111
                                        51111
                                        3111111
                                        111111111

Partitions of odd length are counted by A027193, ranked by A026424.
Partitions with odd maximum are counted by A027193, ranked by A244991.
The Heinz numbers of these partitions are given by A340386.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
A000009 counts partitions into odd parts, ranked by A066208.
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A101707 counts partitions with odd rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A027187, A078408, A174725, A236914.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]*Max[#]]&]],{n,30}]

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

Original entry on oeis.org

2, 5, 8, 11, 17, 20, 23, 30, 31, 32, 41, 44, 45, 47, 50, 59, 66, 67, 68, 73, 75, 80, 83, 92, 97, 99, 102, 103, 109, 110, 120, 124, 125, 127, 128, 137, 138, 149, 153, 154, 157, 164, 165, 167, 170, 176, 179, 180, 186, 188, 191, 197, 200, 207, 211, 227, 230

Offset: 1

Gus Wiseman, Jan 25 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)             59: (17)           120: (3,2,1,1,1)
      5: (3)             66: (5,2,1)        124: (11,1,1)
      8: (1,1,1)         67: (19)           125: (3,3,3)
     11: (5)             68: (7,1,1)        127: (31)
     17: (7)             73: (21)           128: (1,1,1,1,1,1,1)
     20: (3,1,1)         75: (3,3,2)        137: (33)
     23: (9)             80: (3,1,1,1,1)    138: (9,2,1)
     30: (3,2,1)         83: (23)           149: (35)
     31: (11)            92: (9,1,1)        153: (7,2,2)
     32: (1,1,1,1,1)     97: (25)           154: (5,4,1)
     41: (13)            99: (5,2,2)        157: (37)
     44: (5,1,1)        102: (7,2,1)        164: (13,1,1)
     45: (3,2,2)        103: (27)           165: (5,3,2)
     47: (15)           109: (29)           167: (39)
     50: (3,3,1)        110: (5,3,1)        170: (7,3,1)

Note: Heinz numbers are given in parentheses below.
The case of odd length only is A026424.
The case of odd maximum only is A244991.
Positions of odd terms in A326846.
These partitions are counted by A340385.
The version for factorizations is A340607.
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length, or of odd maximum.
A061395 gives maximum prime index.
A106529 lists numbers with Omega equal to maximum prime index.
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
Cf. A001222, A027187, A056239, A112798, A236914, A258116, A300063, A324522, A340608, A340788, A340831.

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

Intersection of A026424 (odd length) and A244991 (odd maximum).

A340692 Number of integer partitions of n of odd rank.

Original entry on oeis.org

0, 0, 2, 0, 4, 2, 8, 4, 14, 12, 26, 22, 44, 44, 76, 78, 126, 138, 206, 228, 330, 378, 524, 602, 814, 950, 1252, 1466, 1900, 2238, 2854, 3362, 4236, 5006, 6232, 7356, 9078, 10720, 13118, 15470, 18800, 22152, 26744, 31456, 37772, 44368, 53002, 62134, 73894

Offset: 0

Gus Wiseman, Jan 29 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots):
  .  .  (2)   .  (4)     (32)   (6)       (52)     (8)         (54)
        (11)     (31)    (221)  (33)      (421)    (53)        (72)
                 (211)          (51)      (3211)   (71)        (432)
                 (1111)         (222)     (22111)  (422)       (441)
                                (411)              (431)       (621)
                                (3111)             (611)       (3222)
                                (21111)            (3221)      (3321)
                                (111111)           (3311)      (5211)
                                                   (5111)      (22221)
                                                   (22211)     (42111)
                                                   (41111)     (321111)
                                                   (311111)    (2211111)
                                                   (2111111)
                                                   (11111111)

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.
The case of length/maximum instead of rank is A027193 (A026424/A244991).
The case of odd positive rank is A101707 is (A340604).
The strict case is A117193.
The even version is A340601 (A340602).
The Heinz numbers of these partitions are (A340603).
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.
A064173 counts partitions of positive/negative rank (A340787/A340788).
A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
A101198 counts partitions of rank 1 (A325233).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
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.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A003114, A006141, A027187, A039900, A067538, A096401, A117409, A143773, A324518, A325134, A340828, A340854/A340855.

Table[Length[Select[IntegerPartitions[n],OddQ[Max[#]-Length[#]]&]],{n,0,30}]

Having odd rank is preserved under conjugation, and self-conjugate partitions cannot have odd rank, so a(n) = 2*A101707(n) for n > 0.

A340784 Heinz numbers of even-length integer partitions of even numbers.

Original entry on oeis.org

1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 111, 115, 118, 121, 129, 133, 134, 136, 144, 146, 155, 156, 159, 160, 166, 169, 183, 184, 187, 189, 194, 196, 198, 203, 205, 206, 210, 213, 218, 220

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 28 2024

			The sequence of partitions together with their Heinz numbers begins:
      1: ()            57: (8,2)            118: (17,1)
      4: (1,1)         62: (11,1)           121: (5,5)
      9: (2,2)         64: (1,1,1,1,1,1)    129: (14,2)
     10: (3,1)         81: (2,2,2,2)        133: (8,4)
     16: (1,1,1,1)     82: (13,1)           134: (19,1)
     21: (4,2)         84: (4,2,1,1)        136: (7,1,1,1)
     22: (5,1)         85: (7,3)            144: (2,2,1,1,1,1)
     25: (3,3)         87: (10,2)           146: (21,1)
     34: (7,1)         88: (5,1,1,1)        155: (11,3)
     36: (2,2,1,1)     90: (3,2,2,1)        156: (6,2,1,1)
     39: (6,2)         91: (6,4)            159: (16,2)
     40: (3,1,1,1)     94: (15,1)           160: (3,1,1,1,1,1)
     46: (9,1)        100: (3,3,1,1)        166: (23,1)
     49: (4,4)        111: (12,2)           169: (6,6)
     55: (5,3)        115: (9,3)            183: (18,2)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of prime powers is A056798.
These partitions are counted by A236913.
The odd version is A160786 (A340931).
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
- Even -
A027187 counts partitions of even length/maximum (A028260/A244990).
A034008 counts compositions of even length.
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.
A340601 counts partitions of even rank (A340602).
A340785 counts factorizations into even factors.
A340786 counts even-length factorizations into even factors.
Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.
Squares (A000290) is a subsequence.
Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).
Positions of even terms in A373381.

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

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
isA340784(n) = A353331(n); \\ Antti Karttunen, Apr 14 2022

Intersection of A028260 and A300061.

A340932 Numbers whose least prime index is odd. Heinz numbers of integer partitions whose last part is odd.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 54, 55, 56, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 80, 82, 83, 84, 85, 86, 88, 90, 92, 94, 95, 96, 97

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 included.

			The sequence of terms together with their prime indices begins:
      2: {1}           24: {1,1,1,2}       46: {1,9}
      4: {1,1}         25: {3,3}           47: {15}
      5: {3}           26: {1,6}           48: {1,1,1,1,2}
      6: {1,2}         28: {1,1,4}         50: {1,3,3}
      8: {1,1,1}       30: {1,2,3}         52: {1,1,6}
     10: {1,3}         31: {11}            54: {1,2,2,2}
     11: {5}           32: {1,1,1,1,1}     55: {3,5}
     12: {1,1,2}       34: {1,7}           56: {1,1,1,4}
     14: {1,4}         35: {3,4}           58: {1,10}
     16: {1,1,1,1}     36: {1,1,2,2}       59: {17}
     17: {7}           38: {1,8}           60: {1,1,2,3}
     18: {1,2,2}       40: {1,1,1,3}       62: {1,11}
     20: {1,1,3}       41: {13}            64: {1,1,1,1,1,1}
     22: {1,5}         42: {1,2,4}         65: {3,6}
     23: {9}           44: {1,1,5}         66: {1,2,5}

These partitions are counted by A026804.
The case where all prime indices are odd is A066208.
Looking at greatest prime index instead of least gives A244991.
Every term x is a product of A257991(x) elements of A341446.
The complement is {1} \/ A340933, counted by A026805.
A001222 counts prime factors.
A005408 lists odd numbers.
A027193 counts odd-length partitions, ranked by A026424.
A031368 lists odd-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058695 counts partitions of odd numbers, ranked by A300063.
A061395 selects greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A006141, A160786, A300272, A340604, A340854, A340855.

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

A055396(a(n)) belongs to A005408.

Closed under multiplication.

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.

A340931 Heinz numbers of integer partitions of odd numbers into an odd number of parts.

Original entry on oeis.org

2, 5, 8, 11, 17, 18, 20, 23, 31, 32, 41, 42, 44, 45, 47, 50, 59, 67, 68, 72, 73, 78, 80, 83, 92, 97, 98, 99, 103, 105, 109, 110, 114, 124, 125, 127, 128, 137, 149, 153, 157, 162, 164, 167, 168, 170, 174, 176, 179, 180, 182, 188, 191, 195, 197, 200, 207, 211

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with the corresponding partitions begins:
      2: (1)             50: (3,3,1)        109: (29)
      5: (3)             59: (17)           110: (5,3,1)
      8: (1,1,1)         67: (19)           114: (8,2,1)
     11: (5)             68: (7,1,1)        124: (11,1,1)
     17: (7)             72: (2,2,1,1,1)    125: (3,3,3)
     18: (2,2,1)         73: (21)           127: (31)
     20: (3,1,1)         78: (6,2,1)        128: (1,1,1,1,1,1,1)
     23: (9)             80: (3,1,1,1,1)    137: (33)
     31: (11)            83: (23)           149: (35)
     32: (1,1,1,1,1)     92: (9,1,1)        153: (7,2,2)
     41: (13)            97: (25)           157: (37)
     42: (4,2,1)         98: (4,4,1)        162: (2,2,2,2,1)
     44: (5,1,1)         99: (5,2,2)        164: (13,1,1)
     45: (3,2,2)        103: (27)           167: (39)
     47: (15)           105: (4,3,2)        168: (4,2,1,1,1)

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A160786.
The even version is A236913 (A340784).
The case of where the prime indices are also odd is A300272.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts odd-length partitions (A026424).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
Cf. A000041, A316413, A326845, A340385, A340386, A340387, A340604, A340607, A340854, A340855.

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

Intersection of A026424 and A300063.

cp's OEIS Frontend

A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A340692 Number of integer partitions of n of odd rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A340784 Heinz numbers of even-length integer partitions of even numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A340932 Numbers whose least prime index is odd. Heinz numbers of integer partitions whose last part is odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs