A327482 -id:A327482 - cp's OEIS frontend

A359904 Numbers whose prime factors and prime signature have the same mean.

1, 4, 27, 400, 3125, 9072, 10800, 14580, 24057, 35721, 50625, 73984, 117760, 134400, 158976, 181440, 191488, 389376, 452709, 544000, 583680, 664848, 731136, 774400, 823543, 878592, 965888

Offset: 1

Gus Wiseman, Jan 25 2023

nonn

The multiset of prime factors of n is row n of A027746.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime factors begin:
      1: {}
      4: {2,2}
     27: {3,3,3}
    400: {2,2,2,2,5,5}
   3125: {5,5,5,5,5}
   9072: {2,2,2,2,3,3,3,3,7}
  10800: {2,2,2,2,3,3,3,5,5}
  14580: {2,2,3,3,3,3,3,3,5}
  24057: {3,3,3,3,3,3,3,11}
  35721: {3,3,3,3,3,3,7,7}
  50625: {3,3,3,3,5,5,5,5}
  73984: {2,2,2,2,2,2,2,2,17,17}

The prime factors are A027746, mean A123528/A123529.
The prime signature is A124010, mean A088529/A088530.
For prime indices instead of factors we have A359903.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A078175 = numbers whose prime factors have integer mean, indices A316413.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A360005 gives median of prime indices (times two).
Cf. A240219, A326622, A327473, A359905, A359908, A359913, A360008, A360068.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]

A360952 Number of strict integer partitions of n with non-integer median; a(0) = 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 6, 1, 8, 4, 11, 5, 15, 10, 20, 13, 27, 22, 36, 28, 47, 43, 63, 56, 82, 79, 107, 103, 140, 141, 180, 181, 232, 242, 299, 308, 380, 402, 483, 511, 613, 656, 772, 824, 969, 1047, 1215, 1309, 1514, 1642, 1882, 2039, 2334, 2539, 2882

Offset: 0

Gus Wiseman, Mar 10 2023

nonn

All of these partitions have even length.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(0) = 1 through a(15) = 11 partitions (0 = {}, A..E = 10..14):
  0  .  .  21  .  32  .  43  .  54  4321  65    6321  76    5432  87
                  41     52     63        74          85    6431  96
                         61     72        83          94    6521  A5
                                81        92          A3    8321  B4
                                          A1          B2          C3
                                          5321        C1          D2
                                                      5431        E1
                                                      7321        6432
                                                                  7431
                                                                  7521
                                                                  9321

The non-strict version is A307683, ranks A359912.
The non-strict complement is A325347, ranks A359908.
The strict complement is counted by A359907.
For mean instead of median we have A361391, non-strict A349156.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 = partitions with integer mean, complement A102627, ranks A316413.
A359893/A359901/A359902 count partitions by median.
A360005(n)/2 ranks the median statistic.
Cf. A000975, A051293, A082550, A240219, A240850, A327475, A359897.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!IntegerQ[Median[#]]&]],{n,0,30}]

a(n) = A000009(n) - A359907(n).

A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

Original entry on oeis.org

0, 1, 1, 4, 5, 8, 12, 19, 25, 38, 51, 70, 93, 124, 162, 217, 279, 360, 462, 601, 750, 955, 1203, 1502, 1881, 2336, 2892, 3596, 4407, 5416, 6623, 8083, 9830, 11943, 14471, 17488, 21059, 25317, 30376, 36424, 43489, 51906, 61789, 73498, 87186, 103253, 122098

Offset: 1

Gus Wiseman, Apr 28 2023

nonn

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

			The weighted sum of y = (3,3,1) is 1*3+2*3+3*1 = 12, which is not a multiple of 7, so y is counted under a(7).
The a(2) = 1 through a(7) = 12 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)
              (31)    (41)    (42)      (52)
              (211)   (221)   (51)      (61)
              (1111)  (311)   (321)     (322)
                      (2111)  (411)     (331)
                              (2211)    (421)
                              (21111)   (511)
                              (111111)  (2221)
                                        (4111)
                                        (22111)
                                        (31111)
                                        (211111)

For median instead of mean we have A322439 aerated, complement A362558.
The complement is counted by A362559.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A327483 Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 22, 34, 8, 1, 1, 231, 919, 249, 16, 1, 1, 8349, 112540, 55974, 1906, 32, 1, 1, 1741630, 107608848, 161410965, 4602893, 14905, 64, 1, 1, 4351078600, 1949696350591, 12623411092535, 676491536028, 461346215, 117874, 128, 1

Offset: 0

Gus Wiseman, Sep 13 2019

T(n,k) is the number of partitions of 2^n into 2^(n-k) parts. - Chai Wah Wu, Sep 21 2023

			Triangle begins:
      1
      1       1
      1       2         1
      1       5         4         1
      1      22        34         8       1
      1     231       919       249      16     1
      1    8349    112540     55974    1906    32  1
      1 1741630 107608848 161410965 4602893 14905 64 1
      ...

Alois P. Heinz, Rows n = 0..13, flattened

Row sums are A327484.
Column k = 1 is A068413 (shifted once to the right).
Cf. A067538, A237984, A240850, A327481, A327482.

Table[Length[Select[IntegerPartitions[2^n],Mean[#]==2^k&]],{n,0,5},{k,0,n}]

from sympy.utilities.iterables import partitions
from sympy import npartitions
def A327483_T(n,k):
    if k == 0 or k == n: return 1
    if k == n-1: return 1<Chai Wah Wu, Sep 21 2023

# uses A008284_T
def A327483_T(n,k): return A008284_T(1<Chai Wah Wu, Sep 21 2023

T(n+1,n) = 2^n for n >= 0. - Chai Wah Wu, Sep 14 2019

a(28)-a(35) from Chai Wah Wu, Sep 14 2019

Row n=8 from Alois P. Heinz, Sep 21 2023

A363745 Number of integer partitions of n whose rounded-down mean is 2.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 4, 10, 6, 16, 21, 24, 32, 58, 47, 85, 111, 119, 158, 248, 217, 341, 442, 461, 596, 867, 792, 1151, 1465, 1506, 1916, 2652, 2477, 3423, 4298, 4381, 5488, 7334, 6956, 9280, 11503, 11663, 14429, 18781, 17992, 23383, 28675, 28970, 35449, 45203

Offset: 0

Gus Wiseman, Jul 05 2023

nonn

			The a(2) = 1 through a(10) = 16 partitions:
  (2)  .  (22)  (32)  (222)  (322)  (332)   (3222)  (3322)
          (31)  (41)  (321)  (331)  (422)   (3321)  (3331)
                      (411)  (421)  (431)   (4221)  (4222)
                             (511)  (521)   (4311)  (4321)
                                    (611)   (5211)  (4411)
                                    (2222)  (6111)  (5221)
                                    (3221)          (5311)
                                    (3311)          (6211)
                                    (4211)          (7111)
                                    (5111)          (22222)
                                                    (32221)
                                                    (33211)
                                                    (42211)
                                                    (43111)
                                                    (52111)
                                                    (61111)

For 1 instead of 2 we have A025065, ranks A363949.
The high version is A026905 reduplicated, ranks A363950.
Column k = 2 of A363945.
These partitions have ranks A363954.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A027336, A237984, A241131, A327472, A327482, A363723, A363943, A363944, A363946.

Table[Length[Select[IntegerPartitions[n],Floor[Mean[#]]==2&]],{n,0,30}]

A327484 Number of integer partitions of 2^n whose mean is a power of 2.

Original entry on oeis.org

1, 2, 4, 11, 66, 1417, 178803, 275379307, 15254411521973, 108800468645440803267, 964567296140908420613296779144, 219614169629364529542990295052656098001967511, 38626966436500261962963100479469496821891576834974275502742922521

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

Number of partitions of 2^n whose number of parts is a power of 2. - Chai Wah Wu, Sep 21 2023

			The a(0) = 1 through a(3) = 11 partitions:
  (1)  (2)   (4)     (8)
       (11)  (22)    (44)
             (31)    (53)
             (1111)  (62)
                     (71)
                     (2222)
                     (3221)
                     (3311)
                     (4211)
                     (5111)
                     (11111111)

Chai Wah Wu, Table of n, a(n) for n = 0..15 (n = 0..13 from Alois P. Heinz)

Row sums of A327483.

Cf. A067538, A068413, A135342, A237984, A327474, A327481, A327482.

Table[Length[Select[IntegerPartitions[2^n],IntegerQ[Mean[#]]&]],{n,0,5}]

from sympy.utilities.iterables import partitions
def A327484(n): return sum(1 for s,p in partitions(1<Chai Wah Wu, Sep 21 2023

# uses A008284_T
def A327484(n): return sum(A008284_T(1<Chai Wah Wu, Sep 21 2023

a(7) from Chai Wah Wu, Sep 14 2019
a(8)-a(11) from Alois P. Heinz, Sep 21 2023
a(12) from Chai Wah Wu, Sep 21 2023

A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

Original entry on oeis.org

10, 39, 68, 115, 138, 259, 310, 328, 387, 517, 574, 636, 793, 795, 1034, 1168, 1206, 1241, 1281, 1340, 1534, 1691, 1825, 2212, 2278, 2328, 2343, 2369, 2370, 2727, 2774, 2905, 3081, 3277, 3818, 3924, 4064, 4074, 4247, 4268, 4360, 4539, 4850, 4905, 5243, 5335

Offset: 1

Gus Wiseman, Feb 18 2023

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 terms together with their prime indices begin:
     1: {}
    10: {1,3}
    39: {2,6}
    68: {1,1,7}
   115: {3,9}
   138: {1,2,9}
   259: {4,12}
   310: {1,3,11}
   328: {1,1,1,13}
   387: {2,2,14}
   517: {5,15}
   574: {1,4,13}
   636: {1,1,2,16}
For example, the prime indices of 138 are {1,2,9}, with mean 4, and with first differences (1,7), with mean also 4, so 138 is in the sequence.

These partitions are counted by A360670.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A301987 lists numbers whose sum of prime indices equals their product.
A316413 lists numbers whose prime indices have integer mean.
A334201 adds up all prime indices except the greatest.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A344415, A348551, A359904, A360008, A360068, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Mean[prix[#]]==Mean[Differences[prix[#]]]&]

A361655 Number of even-length integer partitions of 2n with integer mean.

Original entry on oeis.org

0, 1, 3, 4, 10, 6, 33, 8, 65, 68, 117, 12, 583, 14, 319, 1078, 1416, 18, 3341, 20, 8035, 5799, 1657, 24, 36708, 16954, 3496, 24553, 68528, 30, 192180, 32, 178802, 91561, 14625, 485598, 955142, 38, 29223, 316085, 2622697, 42, 3528870, 44, 2443527, 5740043

Offset: 0

Gus Wiseman, Mar 23 2023

nonn

			The a(0) = 0 through a(5) = 6 partitions:
  .  (11)  (22)    (33)      (44)        (55)
           (31)    (42)      (53)        (64)
           (1111)  (51)      (62)        (73)
                   (111111)  (71)        (82)
                             (2222)      (91)
                             (3221)      (1111111111)
                             (3311)
                             (4211)
                             (5111)
                             (11111111)
For example, the partition (4,2,1,1) has length 4 and mean 2, so is counted under a(4).

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

Even-length partitions are counted by A027187, bisection A236913.
Including odd-length partitions gives A067538 bisected, ranks A316413.
For median instead of mean we have A361653.
The odd-length version is counted by A361656.
A000041 counts integer partitions, strict A000009.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A326622 counts factorizations with integer mean.
Cf. A000016, A027193, A067659, A082550, A102627, A237984, A240219, A327475, A348551, A359893.

Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,15}]

a(n)=if(n==0, 0, sumdiv(n, d, polcoef(1/prod(k=1, 2*d, 1 - x^k + O(x*x^(2*(n-d)))), 2*(n-d)))) \\ Andrew Howroyd, Mar 24 2023

Terms a(36) and beyond from Andrew Howroyd, Mar 24 2023

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353

Offset: 0

Gus Wiseman, May 23 2023

nonn

Equivalently, n = (length)*(minimum).

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

A363134 Positive integers whose multiset of prime indices satisfies: (length) = 2*(minimum).

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 81, 82, 86, 94, 106, 118, 122, 134, 135, 142, 146, 158, 166, 178, 189, 194, 202, 206, 214, 218, 225, 226, 254, 262, 274, 278, 297, 298, 302, 314, 315, 326, 334, 346, 351, 358, 362, 375, 382, 386, 394, 398, 422, 441

Offset: 1

Gus Wiseman, Jun 05 2023

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 terms together with their prime indices begin:
     4: {1,1}         94: {1,15}       214: {1,28}
     6: {1,2}        106: {1,16}       218: {1,29}
    10: {1,3}        118: {1,17}       225: {2,2,3,3}
    14: {1,4}        122: {1,18}       226: {1,30}
    22: {1,5}        134: {1,19}       254: {1,31}
    26: {1,6}        135: {2,2,2,3}    262: {1,32}
    34: {1,7}        142: {1,20}       274: {1,33}
    38: {1,8}        146: {1,21}       278: {1,34}
    46: {1,9}        158: {1,22}       297: {2,2,2,5}
    58: {1,10}       166: {1,23}       298: {1,35}
    62: {1,11}       178: {1,24}       302: {1,36}
    74: {1,12}       189: {2,2,2,4}    314: {1,37}
    81: {2,2,2,2}    194: {1,25}       315: {2,2,3,4}
    82: {1,13}       202: {1,26}       326: {1,38}
    86: {1,14}       206: {1,27}       334: {1,39}

Partitions of this type are counted by A237757.
Removing the factor 2 gives A324522.
For maximum instead of length we have A361908, counted by A118096.
For mean instead of length we have A363133, counted by A363132.
For maximum instead of minimum we have A363218, counted by A237753.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Cf. A000961, A006141, A046660, A051293, A106529, A111907, A237755, A237824, A327482, A361860, A361861, A362050.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[prix[#]]==2*Min[prix[#]]&]

A001222(a(n)) = 2*A055396(a(n)).

cp's OEIS Frontend

A359904 Numbers whose prime factors and prime signature have the same mean.

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A360952 Number of strict integer partitions of n with non-integer median; a(0) = 1.

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A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

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A327483 Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.

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A363745 Number of integer partitions of n whose rounded-down mean is 2.

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A327484 Number of integer partitions of 2^n whose mean is a power of 2.

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A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

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A361655 Number of even-length integer partitions of 2n with integer mean.

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