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 1-5 of 5 results.

A327778 Number of integer partitions of n whose LCM is a multiple of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 11, 1, 11, 23, 1, 1, 23, 1, 85, 85, 45, 1, 152, 1, 84, 1, 451, 1, 1787, 1, 1, 735, 260, 1925, 1908, 1, 437, 1877, 4623, 1, 14630, 1, 6934, 10519, 1152, 1, 6791, 1, 1817, 10159, 22556, 1, 2819, 47927, 69333, 22010, 4310, 1
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2019

Keywords

Examples

			The partitions of n = 6, 10, 12, and 15 whose LCM is a multiple of n:
  (6)      (10)         (12)             (15)
  (3,2,1)  (5,3,2)      (5,4,3)          (6,5,4)
           (5,4,1)      (6,4,2)          (7,5,3)
           (5,2,2,1)    (8,3,1)          (9,5,1)
           (5,2,1,1,1)  (4,3,3,2)        (10,3,2)
                        (4,4,3,1)        (5,4,3,3)
                        (6,4,1,1)        (5,5,3,2)
                        (4,3,2,2,1)      (6,5,2,2)
                        (4,3,3,1,1)      (6,5,3,1)
                        (4,3,2,1,1,1)    (10,3,1,1)
                        (4,3,1,1,1,1,1)  (5,3,3,2,2)
                                         (5,3,3,3,1)
                                         (5,4,3,2,1)
                                         (5,5,3,1,1)
                                         (6,5,2,1,1)
                                         (5,3,2,2,2,1)
                                         (5,3,3,2,1,1)
                                         (5,4,3,1,1,1)
                                         (6,5,1,1,1,1)
                                         (5,3,2,2,1,1,1)
                                         (5,3,3,1,1,1,1)
                                         (5,3,2,1,1,1,1,1)
                                         (5,3,1,1,1,1,1,1,1)

Links

Crossrefs

The Heinz numbers of these partitions are given by A327783.
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. A000961, A018818, A067538, A290103, A319333, A326842, A326843, A327780.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:=
      proc(n, i, l) option remember; `if`(n=0 or i=1,
        `if`(l=m, 1, 0), `if`(i<2, 0, b(n, i-1, l))+
         b(n-i, min(n-i, i), igcd(m, ilcm(l, i))))
      end; `if`(isprime(m), 1, b(m$2, 1))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 26 2019
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Divisible[LCM@@#,n]&]],{n,30}]
(* Second program: *)
a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] = If[n == 0 || i == 1, If[l == m, 1, 0], If[i<2, 0, b[n, i - 1, l]] + b[n - i, Min[n - i, i], GCD[m, LCM[l, i]]]]; If[PrimeQ[m], 1, b[m, m, 1]]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

Formula

a(n) = 1 <=> n in { A000961 }. - Alois P. Heinz, Sep 26 2019

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}]

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

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)).
Showing 1-5 of 5 results.

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