A324515 -id:A324515 - cp's OEIS frontend

A324522 Numbers > 1 where the minimum prime index is equal to the number of prime factors counted with multiplicity.

2, 9, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 125, 129, 141, 159, 175, 177, 183, 201, 213, 219, 237, 245, 249, 267, 275, 291, 303, 309, 321, 325, 327, 339, 381, 385, 393, 411, 417, 425, 447, 453, 455, 471, 475, 489, 501, 519, 537, 543, 573, 575, 579, 591

Offset: 1

Gus Wiseman, Mar 06 2019

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 where the minimum part is equal to the number of parts (A006141). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
    2: {1}
    9: {2,2}
   15: {2,3}
   21: {2,4}
   33: {2,5}
   39: {2,6}
   51: {2,7}
   57: {2,8}
   69: {2,9}
   87: {2,10}
   93: {2,11}
  111: {2,12}
  123: {2,13}
  125: {3,3,3}
  129: {2,14}
  141: {2,15}
  159: {2,16}
  175: {3,3,4}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001222, A006141, A055396, A056239, A106529, A112798, A256617.

Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(min(factorset(n)))=bigomega(n)):
select(q, [$2..600])[];  # Alois P. Heinz, Mar 07 2019

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

A055396(a(n)) = A001222(a(n)).

A324521 Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 126, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256

Offset: 1

Gus Wiseman, Mar 06 2019

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 with nonnegative rank (A064174). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  27: {2,2,2}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  40: {1,1,1,3}
  45: {2,2,3}
  48: {1,1,1,1,2}

Matthieu Pluntz, Table of n, a(n) for n = 1..10929 (up to a(n) = 2^21)

Cf. A001222, A003114, A056239, A061395, A064174, A106529, A112798, A256617.

Cf. A324515, A324517, A324519, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(max(factorset(n)))<=bigomega(n)):
select(q, [$2..300])[];  # Alois P. Heinz, Mar 07 2019

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

isok(m) = (m>1) && (primepi(vecmax(factor(m)[, 1])) <= bigomega(m)); \\ Michel Marcus, Nov 14 2022

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return primepi(max(f)) <= sum(f.values())
print([k for k in range(2, 257) if ok(k)]) # Michael S. Branicky, Nov 15 2022

A061395(a(n)) <= A001222(a(n)).

A324517 Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

4, 24, 27, 36, 54, 80, 200, 224, 240, 360, 405, 500, 540, 600, 625, 672, 675, 704, 784, 810, 900, 1008, 1120, 1125, 1250, 1350, 1500, 1512, 1664, 1701, 1875, 2112, 2250, 2268, 2352, 2744, 2800, 3168, 3360, 3402, 3520, 3528, 3750, 3872, 3920, 3969, 4352, 4752

Offset: 1

Gus Wiseman, Mar 06 2019

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 the integer partitions enumerated by A324518. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
   24: {1,1,1,2}
   27: {2,2,2}
   36: {1,1,2,2}
   54: {1,2,2,2}
   80: {1,1,1,1,3}
  200: {1,1,1,3,3}
  224: {1,1,1,1,1,4}
  240: {1,1,1,1,2,3}
  360: {1,1,1,2,2,3}
  405: {2,2,2,2,3}
  500: {1,1,3,3,3}
  540: {1,1,2,2,2,3}
  600: {1,1,1,2,3,3}
  625: {3,3,3,3}
  672: {1,1,1,1,1,2,4}
  675: {2,2,2,3,3}
  704: {1,1,1,1,1,1,5}
  784: {1,1,1,1,4,4}
  810: {1,2,2,2,2,3}
  900: {1,1,2,2,3,3}

Cf. A001221, A001222, A046660, A056239, A061395, A112798, A243055, A256617.

Cf. A324515, A324518, A324519, A324521, A324522, A324560, A324562.

Select[Range[2,1000],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]==Total[Last/@f]-Length[f]]&]

A061395(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

A340602 Heinz numbers of integer partitions of even rank.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. 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:
     1: ()           31: (11)           58: (10,1)
     2: (1)          32: (1,1,1,1,1)    59: (17)
     5: (3)          35: (4,3)          65: (6,3)
     6: (2,1)        36: (2,2,1,1)      66: (5,2,1)
     8: (1,1,1)      38: (8,1)          67: (19)
     9: (2,2)        39: (6,2)          68: (7,1,1)
    11: (5)          41: (13)           73: (21)
    14: (4,1)        44: (5,1,1)        74: (12,1)
    17: (7)          45: (3,2,2)        75: (3,3,2)
    20: (3,1,1)      47: (15)           80: (3,1,1,1,1)
    21: (4,2)        49: (4,4)          81: (2,2,2,2)
    23: (9)          50: (3,3,1)        83: (23)
    24: (2,1,1,1)    54: (2,2,2,1)      84: (4,2,1,1)
    26: (6,1)        56: (4,1,1,1)      86: (14,1)
    30: (3,2,1)      57: (8,2)          87: (10,2)

FindStat, St000145: The Dyson rank of a partition

Taking only length gives A001222.
Taking only maximum part gives A061395.
These partitions are counted by A340601.
The complement is A340603.
The case of positive rank is A340605.
- Rank -
A047993 counts partitions of rank 0 (A106529).
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.
A324516 counts partitions with rank = maximum minus minimum part (A324515).
A340653 counts factorizations of rank 0.
A340692 counts partitions of odd rank (A340603).
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.

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

Either n = 1 or A061395(n) - A001222(n) is even.

A324519 Numbers > 1 where the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

4, 12, 18, 20, 27, 28, 44, 50, 52, 60, 68, 76, 84, 90, 92, 98, 116, 124, 126, 132, 135, 140, 148, 150, 156, 164, 172, 188, 189, 198, 204, 212, 220, 225, 228, 234, 236, 242, 244, 260, 268, 276, 284, 292, 294, 297, 306, 308, 316, 332, 338, 340, 342, 348, 350

Offset: 1

Gus Wiseman, Mar 06 2019

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 the integer partitions enumerated by A324520. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   4: {1,1}
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  27: {2,2,2}
  28: {1,1,4}
  44: {1,1,5}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  68: {1,1,7}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}

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

Cf. A001221, A001222, A046660, A055396, A056239, A061395, A106529, A112798.

Cf. A324515, A324517, A324520, A324521, A324522, A324560, A324562.

Select[Range[2,100],With[{f=FactorInteger[#]},PrimePi[f[[1,1]]]==Total[Last/@f]-Length[f]]&]

A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

A324560 Numbers > 1 where the minimum prime index is less than or equal to the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100

Offset: 1

Gus Wiseman, Mar 06 2019

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 a certain type of integer partitions counted by A039900 (but not the type of partitions described in the name). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}

Cf. A001222, A039900, A055396, A056239, A061395, A106529, A112798.

Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(min(factorset(n)))<=bigomega(n)):
select(q, [$2..100])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[2,100],PrimePi[FactorInteger[#][[1,1]]]<=PrimeOmega[#]&]

A055396(a(n)) <= A001222(a(n)).

A324516 Number of integer partitions of n > 0 where the maximum part minus the minimum part equals the length minus the number of distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 5, 2, 8, 6, 6, 10, 14, 12, 20, 27, 23, 40, 40, 51, 62, 82, 88, 123, 135, 173, 197, 253, 285, 350, 419, 497, 594, 708, 855, 978, 1195, 1395, 1648, 1915, 2313, 2625, 3170, 3625, 4336, 4948, 5900, 6751, 7970, 9180, 10704, 12337, 14436, 16517

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324515.

			The a(8) = 5 through a(14) = 14 integer partitions:
  (8)      (9)      (A)       (B)       (C)        (D)        (E)
  (332)    (32211)  (433)     (443)     (4422)     (544)      (554)
  (3311)            (3331)    (33221)   (33321)    (43222)    (4442)
  (32111)           (4222)    (44111)   (422211)   (52222)    (5333)
  (41111)           (32221)   (422111)  (5211111)  (422221)   (43322)
                    (33211)   (431111)  (6111111)  (433111)   (44411)
                    (421111)                       (442111)   (442211)
                    (511111)                       (4321111)  (443111)
                                                   (5221111)  (551111)
                                                   (5311111)  (4322111)
                                                              (5222111)
                                                              (5411111)
                                                              (62111111)
                                                              (71111111)

Cf. A006141, A039900, A046660, A047993, A064174, A243055.

Cf. A324515, A324518, A324520.

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Length[#]-Length[Union[#]]&]],{n,30}]

A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Gus Wiseman, Mar 16 2023

nonn

Differs from A068938 in having 1 and 4 and lacking 80.

Includes all squarefree numbers.

			The prime indices of 80 are {1,1,1,1,3}, with 5 parts and 2 distinct parts, and 2*2 < 5, so 80 is not in the sequence.

Complement of A360558.
Positions of nonnegative terms in A361205.
These partitions are counted by A361394.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324521, A324560, A324562.
Cf. A360248, A360249, A360454.

Select[Range[100],2*PrimeNu[#]>=PrimeOmega[#]&]

A001222(a(n)) <= 2*A001221(a(n)).

A361393 Positive integers k such that 2*omega(k) > bigomega(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Mar 16 2023

nonn

First differs from A317090 in having 120 and lacking 360.

There are numbers like 1, 120, 168, 180, 252,... which are not in A179983 but in here, and others like 360, 504, 540, 600,... which are in A179983 but not in here. - R. J. Mathar, Mar 21 2023

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
The prime indices of 120 are {1,1,1,2,3}, with 3 distinct parts and 5 parts, and 2*3 > 5, so 120 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with 3 distinct parts and 6 parts, and 2*3 is not greater than 6, so 360 is not in the sequence.

These partitions are counted by A237365.
The complement is A361204.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

isA361393 := proc(n)
    if 2*A001221(n) > numtheory[bigomega](n) then
        true;
    else
        false ;
    end if:
end proc:
for n from 1 to 100 do
    if isA361393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023

Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]

{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023

A340929 Heinz numbers of integer partitions of odd negative rank.

Original entry on oeis.org

4, 12, 16, 18, 27, 40, 48, 60, 64, 72, 90, 100, 108, 112, 135, 150, 160, 162, 168, 192, 225, 240, 243, 250, 252, 256, 280, 288, 352, 360, 375, 378, 392, 400, 420, 432, 448, 528, 540, 567, 588, 600, 625, 630, 640, 648, 672, 700, 768, 792, 810, 832, 880, 882

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)             150: (3,3,2,1)
      12: (2,1,1)           160: (3,1,1,1,1,1)
      16: (1,1,1,1)         162: (2,2,2,2,1)
      18: (2,2,1)           168: (4,2,1,1,1)
      27: (2,2,2)           192: (2,1,1,1,1,1,1)
      40: (3,1,1,1)         225: (3,3,2,2)
      48: (2,1,1,1,1)       240: (3,2,1,1,1,1)
      60: (3,2,1,1)         243: (2,2,2,2,2)
      64: (1,1,1,1,1,1)     250: (3,3,3,1)
      72: (2,2,1,1,1)       252: (4,2,2,1,1)
      90: (3,2,2,1)         256: (1,1,1,1,1,1,1,1)
     100: (3,3,1,1)         280: (4,3,1,1,1)
     108: (2,2,2,1,1)       288: (2,2,1,1,1,1,1)
     112: (4,1,1,1,1)       352: (5,1,1,1,1,1)
     135: (3,2,2,2)         360: (3,2,2,1,1,1)

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 A101707.
The positive version is A101707 (A340604).
The even version is A101708 (A340930).
The not necessarily odd version is A064173 (A340788).
A001222 counts prime factors.
A027193 counts partitions of odd length (A026424).
A047993 counts balanced partitions (A106529).
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- 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.
A324516 counts partitions with rank equal to maximum minus minimum part (A324515).
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).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&]

For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.

cp's OEIS Frontend

A324522 Numbers > 1 where the minimum prime index is equal to the number of prime factors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A324521 Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A324517 Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A340602 Heinz numbers of integer partitions of even rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A324519 Numbers > 1 where the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A324560 Numbers > 1 where the minimum prime index is less than or equal to the number of prime factors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A324516 Number of integer partitions of n > 0 where the maximum part minus the minimum part equals the length minus the number of distinct parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A361395 Positive integers k such that 2*omega(k) >= bigomega(k).