A319169 -id:A319169 - cp's OEIS frontend

A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 3, 6, 5, 5, 5, 6, 8, 8, 9, 8, 11, 8, 12, 13, 16, 14, 13, 16, 21, 18, 21, 25, 22, 24, 27, 35, 33, 33, 32, 37, 42, 47, 48, 48, 52, 51, 59, 70, 68, 65, 69, 80, 87, 90, 103, 100, 96, 103, 123, 128, 135, 136, 132, 153

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict and its parts must also be squarefree.

			The a(26) = 11 integer partitions:
  (26),
  (15,11), (19,7), (21,5), (23,3),
  (13,7,6), (13,10,3), (13,11,2), (17,7,2), (19,5,2),
  (11,7,5,3).

First differences of A322526.

Cf. A002865, A003963, A005117, A064573, A072774, A073576, A302505, A319005, A319057, A319169, A320322, A322527, A322528, A322529.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&SquareFreeQ[Times@@#]&]],{n,30}]

A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 7, 9, 10, 10, 10, 9, 11, 15, 14, 13, 15, 14, 14, 17, 16, 17, 17, 16, 16, 17, 17, 21, 26, 24, 23, 25, 27, 29, 32, 31, 29, 36, 36, 35, 37, 37, 42, 49, 45, 44, 50, 49, 50, 58, 55, 55, 58, 56, 58, 66, 62, 65, 75

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(15) = 8 partitions are: (15), (14,1), (12,3), (12,2,1), (10,5), (10,4,1), (6,9), (8,6,1).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

Counting prime factors with multiplicity gives A358901.
The weakly decreasing version is A358902, with multiplicity A358335.
A001222 counts prime factors, distinct A001221.
A116608 counts partitions by sum and number of distinct parts.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A071625, A129519, A141199, A319169, A358831, A358909, A358911.

p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t b(n$2):
seq(a(n), n=0..68);  # Alois P. Heinz, Feb 14 2024

Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeNu/@#&]],{n,0,30}]

a(56) and beyond from Lucas A. Brown, Dec 14 2022

A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075

Offset: 0

Gus Wiseman, Aug 04 2024

nonn

			The a(0) = 1 through a(4) = 15 ways:
  ()  ((1))  ((2))      ((3))          ((4))
             ((1,1))    ((1,2))        ((1,3))
             ((1),(1))  ((1,1,1))      ((2,2))
                        ((1),(1,1))    ((1,1,2))
                        ((1,1),(1))    ((2),(2))
                        ((1),(1),(1))  ((1,1,1,1))
                                       ((1),(1,2))
                                       ((1,2),(1))
                                       ((1),(1,1,1))
                                       ((1,1),(1,1))
                                       ((1,1,1),(1))
                                       ((1),(1),(1,1))
                                       ((1),(1,1),(1))
                                       ((1,1),(1),(1))
                                       ((1),(1),(1),(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A variation for weakly increasing lengths is A141199.
For identical sums instead of minima we have A279787.
The case of reversed twice-partitions is A306319, distinct A358830.
For maxima instead of minima, or for unreversed partitions, we have A358905.
The strict case is A374686 (ranks A374685), maxima A374760 (ranks A374759).
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A063834, A106356, A189076, A238343, A304969, A305551, A319066, A323429, A333213, A358833, A358835.

Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]

seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024

G.f.: 1 + Sum_{k>=1} (-1 + 1/(1 - x^k/Product_{j>=k} (1 - x^j))). - Andrew Howroyd, Dec 29 2024

a(16) onwards from Andrew Howroyd, Dec 29 2024

A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1

Offset: 1

Gus Wiseman, Dec 17 2018

			Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Andrew Howroyd, Table of n, a(n) for n = 1..300

Row sums are A322555.

Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 68, 100, 153, 227, 342, 509, 759, 1129, 1678, 2492, 3699, 5477, 8121, 12015, 17795, 26313, 38924, 57541, 85065, 125712, 185758, 274431, 405420, 598815, 884465, 1306165, 1928943, 2848360, 4205979, 6210289, 9169540

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(0) = 1 through a(6) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (33)
                 (111)  (31)    (32)     (42)
                        (211)   (41)     (51)
                        (1111)  (221)    (222)
                                (311)    (231)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358901.
The version not counting multiplicity is A358902, strict A358903.
The case of partitions is A358909, complement A358910.
The case of equality is A358911, partitions A319169.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (52)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (2222)
                                                          (11111111)

Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SameQ@@Last/@FactorInteger[Times@@#]]&]],{n,30}]

More terms from Alois P. Heinz, Dec 14 2018

A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

			The a(9) = 1 through a(14) = 11 partitions:
  (54)  (541)  (74)    (543)    (76)      (554)
               (542)   (741)    (544)     (743)
               (5411)  (5421)   (742)     (761)
                       (54111)  (5422)    (5432)
                                (5431)    (5441)
                                (7411)    (7421)
                                (54211)   (54221)
                                (541111)  (54311)
                                          (74111)
                                          (542111)
                                          (5411111)

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The complement is counted by A358909.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358902, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],!GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 13, 15, 16, 17, 29, 31, 32, 33, 41, 43, 47, 51, 55, 59, 64, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 128, 137, 139, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 233, 241, 249, 255, 256, 257, 269, 271

Offset: 1

Gus Wiseman, Dec 14 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
All entries are themselves squarefree numbers (except the powers of 2).
The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

			The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

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

A339511 Number of subsets of {1..n} whose elements have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 20, 21, 25, 33, 49, 51, 83, 99, 131, 132, 196, 200, 328, 336, 400, 528, 784, 786, 1042, 1554, 1570, 1602, 2114, 2178, 3202, 3203, 4227, 6275, 10371, 10375, 12423, 20615, 36999, 37007, 41103, 41231, 49423, 49679, 50191, 82959, 99343, 99345, 164881, 165905, 296977, 299025, 331793, 331809, 593953, 593985, 1118273, 2166849, 2232385

Offset: 0

Ilya Gutkovskiy, Dec 07 2020

nonn

			a(6) = 12 subsets: {}, {1}, {2}, {3}, {4}, {5}, {6}, {2, 3}, {2, 5}, {3, 5}, {4, 6} and {2, 3, 5}.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Factor

Cf. A001222, A319169, A339512, A339514.

Array[Count[Subsets@ Range[#], ?(SameQ @@ PrimeOmega[#] &)] &, 16, 0] (* _Michael De Vlieger, Feb 24 2021 *)

from sympy import primeomega
def test(n):
    if n<2: return n-1
    return primeomega(n)
def a(n):
    tests = [test(i) for i in range(n+1)]
    return sum(2**tests.count(v)-1 for v in set(tests))
print([a(n) for n in range(60)]) # Michael S. Branicky, Dec 07 2020

a(n) = 1 + Sum_{k=1..n} 2^A335097(k). - Sebastian Karlsson, Feb 23 2021

a(25)-a(48) from Michael S. Branicky, Dec 07 2020

cp's OEIS Frontend

A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

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A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

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A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

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A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

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A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

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Mathematica

Extensions

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

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Mathematica

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

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A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

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A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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