cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752
Offset: 0

Views

Author

Gus Wiseman, Nov 14 2021

Keywords

Comments

Equivalently, partitions whose length does not divide their sum.
By conjugation, also the number of integer partitions of n with greatest part not dividing n.

Examples

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Crossrefs

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

Formula

a(n > 0) = A000041(n) - A067538(n).

A327779 Number of integer partitions of n whose LCM is greater than n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 3, 7, 9, 18, 16, 31, 42, 61, 87, 133, 169, 246, 302, 411, 545, 738, 874, 1167, 1497, 1945, 2421, 3110, 3498, 4476, 5615, 7061, 8777, 10925, 12957, 16036, 19644, 24061, 28858, 35177, 41572, 50424, 60643, 72953, 87499, 104893, 123821, 147776
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2019

Keywords

Examples

			The a(5) = 1 through a(12) = 16 partitions (empty columns not shown):
  (32)  (43)  (53)   (54)    (64)     (65)      (75)
        (52)  (431)  (72)    (73)     (74)      (543)
              (521)  (432)   (433)    (83)      (651)
                     (522)   (532)    (92)      (732)
                     (531)   (541)    (443)     (741)
                     (4311)  (721)    (533)     (831)
                     (5211)  (4321)   (542)     (921)
                             (5311)   (641)     (5322)
                             (43111)  (722)     (5331)
                                      (731)     (5421)
                                      (4322)    (7221)
                                      (4331)    (7311)
                                      (5321)    (53211)
                                      (5411)    (54111)
                                      (7211)    (72111)
                                      (43211)   (531111)
                                      (53111)
                                      (431111)

Crossrefs

The Heinz numbers of these partitions are given by A327784.
Partitions whose LCM is a multiple of their sum are A327778.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is less than their sum are A327781.
Cf. A018818, A067538, A290103, A326842, A327780, A327783.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],LCM@@#>n&]],{n,30}]

A327781 Number of integer partitions of n whose LCM is less than n.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 9, 12, 18, 22, 30, 37, 52, 69, 89, 110, 143, 163, 204, 243, 298, 374, 451, 516, 620, 790, 932, 1064, 1243, 1454, 1699, 2365, 2733, 3071, 3524, 3945, 4526, 5600, 6361, 7111, 8057, 9405, 10621, 12836, 14395, 16066, 18047, 19860, 22143, 25748
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2019

Keywords

Examples

			The a(2) = 1 through a(8) = 18 partitions:
  (11)  (21)   (22)    (41)     (33)      (61)       (44)
        (111)  (31)    (221)    (42)      (322)      (62)
               (211)   (311)    (51)      (331)      (71)
               (1111)  (2111)   (222)     (421)      (332)
                       (11111)  (411)     (511)      (422)
                                (2211)    (2221)     (611)
                                (3111)    (3211)     (2222)
                                (21111)   (4111)     (3221)
                                (111111)  (22111)    (3311)
                                          (31111)    (4211)
                                          (211111)   (5111)
                                          (1111111)  (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

Links

Crossrefs

The Heinz numbers of these partitions are given by A327776.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is greater than their sum are A327779.
Cf. A018818, A290103, A316413, A319333, A326842, A327778, A327780.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:=
      proc(n, i, l) option remember; `if`(n=0, 1,
       `if`(i>1, b(n, i-1, l), 0) +(h-> `if`(h0, b(m$2, 1), 0)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 10 2019
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],LCM@@#1, b[n, i - 1, l], 0] + Function[h, If[h0, b[m, m, 1], 0]];
a /@ Range[0, 70] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)
  • PARI
    b(m,n)={my(d=divisors(m)); polcoef(1/prod(i=1, #d, 1 - x^d[i] + O(x*x^n)), n)}
a(n)={sum(m=1, n-1, b(m, n)*sum(i=1, (n-1)\m, moebius(i)))} \\ Andrew Howroyd, Oct 09 2019

A327783 Heinz numbers of integer partitions whose LCM is a multiple of their sum.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 181, 190, 191, 193, 197, 198, 199, 211, 223, 227, 229, 233, 239, 241
Offset: 1

Views

Author

Gus Wiseman, Sep 25 2019

Keywords

Comments

First differs from A319333 in having 154.
First nonsquarefree term is 198.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   53: {16}
   59: {17}
   61: {18}
   67: {19}

Crossrefs

The enumeration of these partitions by sum is A327778.
Heinz numbers of partitions whose LCM is twice their sum are A327775.
Heinz numbers of partitions whose LCM is less than their sum are A327776.
Heinz numbers of partitions whose LCM is greater than their sum are A327784.
Cf. A056239, A074761, A112798, A290103, A316413, A326841, A327779.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Divisible[LCM@@primeMS[#],Total[primeMS[#]]]&]

Formula

A056239(a(k)) | A290103(a(k)).

A327775 Heinz numbers of integer partitions whose LCM is twice their sum.

Original entry on oeis.org

154, 190, 435, 580, 714, 952, 1118, 1287, 1430, 1653, 1716, 1815, 1935, 2067, 2150, 2204, 2254, 2288, 2415, 2475, 2580, 2756, 2898, 2970, 3220, 3300, 3440, 3710, 3864, 3960, 3975, 4770, 5152, 5280, 5300, 6360, 6461, 6897, 7514, 8307, 8480, 8619, 8695, 8778
Offset: 1

Views

Author

Gus Wiseman, Sep 25 2019

Keywords

Comments

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

Examples

			The sequence of terms together with their prime indices begins:
   154: {1,4,5}
   190: {1,3,8}
   435: {2,3,10}
   580: {1,1,3,10}
   714: {1,2,4,7}
   952: {1,1,1,4,7}
  1118: {1,6,14}
  1287: {2,2,5,6}
  1430: {1,3,5,6}
  1653: {2,8,10}
  1716: {1,1,2,5,6}
  1815: {2,3,5,5}
  1935: {2,2,3,14}
  2067: {2,6,16}
  2150: {1,3,3,14}
  2204: {1,1,8,10}
  2254: {1,4,4,9}
  2288: {1,1,1,1,5,6}
  2415: {2,3,4,9}
  2475: {2,2,3,3,5}

Links

Crossrefs

The enumeration of these partitions by sum is A327780.
Heinz numbers of partitions whose LCM is less than their sum are A327776.
Heinz numbers of partitions whose LCM is a multiple their sum are A327783.
Heinz numbers of partitions whose LCM is greater than their sum are A327784.
Cf. A056239, A074761, A112798, A290103, A326841 , A327778.

Programs

  • Maple
    q:= n-> (l-> is(ilcm(l[])=2*add(j, j=l)))(map(i->
        numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..10000])[];  # Alois P. Heinz, Sep 27 2019
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],LCM@@primeMS[#]==2*Total[primeMS[#]]&]

Formula

A290103(a(k)) = 2 * A056239(a(k)).

A327780 Number of integer partitions of n whose LCM is 2 * n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 12, 0, 0, 6, 0, 10, 32, 6, 0, 8, 0, 9, 0, 32, 0, 505, 0, 0, 108, 16, 147, 258, 0, 20, 170, 134, 0, 2030, 0, 140, 1865, 30, 0, 80, 0, 105, 350, 236, 0, 419, 500, 617, 474, 49, 0, 40966, 0, 56, 8225, 0, 785
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2019

Keywords

Examples

			The a(10) = 1 through a(20) = 10 partitions (A = 10) (empty columns not shown):
  (541)  (831)  (7421)   (A32)       (9432)     (A82)
                (74111)  (5532)      (9441)     (8552)
                         (6522)      (94221)    (A811)
                         (6531)      (94311)    (85421)
                         (A311)      (942111)   (85511)
                         (53322)     (9411111)  (852221)
                         (65211)                (854111)
                         (532221)               (8522111)
                         (533211)               (85211111)
                         (651111)               (851111111)
                         (5322111)
                         (53211111)

Links

Crossrefs

The Heinz numbers of these partitions are given by A327775.
Partitions whose LCM is a multiple of their sum are A327778.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is greater than their sum are A327779.
Partitions whose LCM is less than their sum are A327781.
Cf. A018818, A290103, A316413, A326842.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],LCM@@#==2*n&]],{n,30}]
  • PARI
    b(m,n)={my(d=divisors(m)); polcoef(1/prod(i=1, #d, 1 - x^d[i] + O(x*x^n)), n)}
a(n)={if(n<1, 0, sumdiv(2*n, d, moebius(d)*b(2*n/d, n)))} \\ Andrew Howroyd, Oct 09 2019

Formula

a(n) = Sum_{d|2*n} mu(d)*([x^n] B(2*n/d, x)) for n > 0, where B(m,x) = 1/(Product_{d|m} 1 - x^d). - Andrew Howroyd, Feb 12 2022
Showing 1-6 of 6 results.

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