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.

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A319320 Number of integer partitions of n such that every distinct submultiset has a different LCM.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 12, 15, 17, 20, 22, 24, 25, 31, 35, 39, 40, 48, 51, 55, 64, 73, 77, 85, 92, 104, 115, 126, 136, 147, 157, 176, 198, 211, 234, 246, 269, 294, 326, 350, 375, 403, 443, 475, 526, 560, 600, 650
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Comments

Note that such partitions are necessarily strict.

Examples

			The a(19) = 12 partitions:
  (19),
  (10,9), (11,8), (12,7), (13,6), (14,5), (15,4), (16,3), (17,2),
  (8,6,5), (11,5,3),
  (7,5,4,3).

Crossrefs

Cf. A074761, A108917, A275972, A290103, A316313, A316429, A316431, A319315, A319318, A319327.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@LCM@@@Union[Rest[Subsets[#]]]&]],{n,30}]

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)).

A316440 Number of integer partitions of n such that every submultiset has an integer average.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 2, 7, 5, 8, 2, 13, 2, 10, 10, 14, 2, 20, 2, 17, 15, 14, 2, 32, 3, 16, 22, 25, 2, 40, 2, 27, 30, 20, 4, 58, 2, 22, 40, 40, 2, 64, 2, 40, 53, 26, 2, 93, 3, 30, 64, 54, 2, 94, 4, 58, 78, 32, 2, 138, 2, 34, 96, 75, 10, 131, 2, 76, 111, 48, 2, 192, 2, 40, 138, 99
Offset: 0

Views

Author

Gus Wiseman, Jul 03 2018

Keywords

Examples

			The a(12) = 13 partitions:
  (12),
  (6,6), (7,5), (8,4), (9,3), (10,2), (11,1),
  (4,4,4), (6,4,2), (8,2,2),
  (3,3,3,3),
  (2,2,2,2,2,2),
  (1,1,1,1,1,1,1,1,1,1,1,1).

Links

Crossrefs

Cf. A067538, A074761, A122768, A143773, A237984, A298423, A316313, A316314.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@Mean/@Union[Rest[Subsets[#]]]&]],{n,20}]

Formula

For a prime p, a(p) = 2. - Max Alekseyev, Sep 02 2023

Extensions

a(0) prepended and more terms added by Max Alekseyev, Sep 02 2023

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

A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

Original entry on oeis.org

9, 18, 21, 25, 27, 36, 39, 42, 45, 49, 50, 54, 57, 63, 65, 72, 75, 78, 81, 84, 87, 90, 91, 98, 99, 100, 105, 108, 111, 114, 115, 117, 121, 125, 126, 129, 130, 133, 135, 144, 147, 150, 153, 156, 159, 162, 168, 169, 171, 174, 175, 180, 182, 183, 185, 189, 195
Offset: 1

Views

Author

Gus Wiseman, Jul 03 2018

Keywords

Comments

Also numbers n > 1 such that A290104(n) > 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			Sequence of partitions whose product is greater than their LCM begins: (22), (221), (42), (33), (222), (2211), (62), (421), (322), (44), (331), (2221), (82), (422), (63), (22111), (332), (621), (2222), (4211).

Crossrefs

Cf. A056239, A074761, A143773, A285572, A289508, A290103, A296150, A316429, A316431, A316433, A316437.

Programs

  • Mathematica
    Select[Range[2,300],With[{pms=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@pms/LCM@@pms>1]&]

A327776 Heinz numbers of integer partitions whose LCM is less than their sum.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 68, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 96, 98, 100, 104, 106, 108, 111, 112
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:
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}

Crossrefs

The enumeration of these partitions by sum is A327781.
Heinz numbers of partitions whose LCM is twice their sum are A327775.
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, A316413, A326841, A327779.

Programs

  • Maple
    q:= n-> (l-> is(ilcm(l[])
      numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..120])[];  # Alois P. Heinz, Sep 27 2019
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],LCM@@primeMS[#]

A327784 Heinz numbers of integer partitions whose LCM is greater than their sum.

Original entry on oeis.org

1, 15, 33, 35, 51, 55, 66, 69, 70, 77, 85, 91, 93, 95, 99, 102, 105, 110, 119, 123, 132, 138, 140, 141, 143, 145, 153, 154, 155, 161, 165, 170, 175, 177, 182, 186, 187, 190, 201, 203, 204, 205, 207, 209, 210, 215, 217, 219, 220, 221, 231, 238, 245, 246, 247, 249
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:
    1: {}
   15: {2,3}
   33: {2,5}
   35: {3,4}
   51: {2,7}
   55: {3,5}
   66: {1,2,5}
   69: {2,9}
   70: {1,3,4}
   77: {4,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
   95: {3,8}
   99: {2,2,5}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  119: {4,7}
  123: {2,13}
  132: {1,1,2,5}

Crossrefs

The enumeration of these partitions by sum is A327779.
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 a multiple their sum are A327783.
Cf. A056239, A074761, A112798, A290103, A316413, A326841, A327781.

Programs

  • Maple
    q:= n-> (l-> is(ilcm(l[])>add(j, j=l)))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..250])[];  # Alois P. Heinz, Sep 27 2019
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],LCM@@primeMS[#]>Total[primeMS[#]]&]

Formula

A290103(a(k)) > A056239(a(k)).
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