A319333 -id:A319333 - cp's OEIS frontend

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

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

Gus Wiseman, Sep 25 2019

nonn

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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.

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

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 *)

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

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

Gus Wiseman, Sep 25 2019

nonn

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..500

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.

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

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 *)

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

Gus Wiseman, Sep 25 2019

nonn

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).

			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}

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.

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

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

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.

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 version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

Original entry on oeis.org

2, 1495, 179417, 231133, 727531, 1378583, 1787387, 3744103, 4556993, 7566167, 18977519, 29629391, 30870587, 34174939, 39973571, 53508983, 70946617, 110779141, 138820187, 139681069, 170583017, 225817751, 409219217, 441317981, 493580417, 539462099, 544392433, 712797613, 802903541

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

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

			The corresponding sequence of partitions, whose length is equal to their GCD and whose sum is equal to their LCM: (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4).

Subsequence of A316430.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A290103, A290104, A316431, A316432, A319328, A319330, A319333.

Select[Range[2,10000],With[{m=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And[LCM@@m==Total[m],GCD@@m==Length[m]]]&]

More terms from Max Alekseyev, Jul 25 2024

A319330 Number of integer partitions of n whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

			A list of all such partitions begins (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4), (16,12,12,8), (25,10,5,5,5), (20,15,15,5,5), (20,15,10,10,5).

Cf. A067538, A074761, A143773, A316432, A319328, A319329, A319333.

Table[Length[Select[IntegerPartitions[n],And[LCM@@#==Total[#],GCD@@#==Length[#]]&]],{n,30}]

a(71)-a(105) from Alois P. Heinz, Sep 18 2018

A319334 Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.

Original entry on oeis.org

30, 198, 264, 273, 364, 490, 525, 630, 700, 840, 918, 1120, 1224, 1495, 1632, 1794, 2392, 2420, 2750, 3105, 3450, 3726, 4140, 4263, 4400, 4466, 4921, 4968, 5481, 5520, 5684, 6327, 6624, 7030, 7040, 7308, 8436, 8832, 9744, 11248, 12992, 14079, 14450, 14993

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

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

			The sequence of all non-singleton integer partitions whose sum is equal to their LCM begins: (321), (5221), (52111), (642), (6411), (4431), (4332), (43221), (43311), (432111), (72221), (4311111), (722111), (963), (7211111), (9621).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319333.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],And[!PrimeQ[#],LCM@@primeMS[#]==Total[primeMS[#]]]&]

A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 13, 14, 15, 16, 31, 32, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 127, 128, 136, 138, 139, 141, 142, 143, 162, 163, 168, 170, 171, 173, 174, 175, 177, 181, 182, 183, 184

Offset: 1

Gus Wiseman, Nov 15 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms and corresponding compositions begin:
      0: ()              36: (3,3)           54: (1,2,1,2)
      1: (1)             37: (3,2,1)         55: (1,2,1,1,1)
      2: (2)             38: (3,1,2)         57: (1,1,3,1)
      3: (1,1)           39: (3,1,1,1)       58: (1,1,2,2)
      4: (3)             41: (2,3,1)         59: (1,1,2,1,1)
      7: (1,1,1)         42: (2,2,2)         60: (1,1,1,3)
      8: (4)             43: (2,2,1,1)       61: (1,1,1,2,1)
     10: (2,2)           44: (2,1,3)         62: (1,1,1,1,2)
     11: (2,1,1)         45: (2,1,2,1)       63: (1,1,1,1,1,1)
     13: (1,2,1)         46: (2,1,1,2)       64: (7)
     14: (1,1,2)         47: (2,1,1,1,1)    127: (1,1,1,1,1,1,1)
     15: (1,1,1,1)       50: (1,3,2)        128: (8)
     16: (5)             51: (1,3,1,1)      136: (4,4)
     31: (1,1,1,1,1)     52: (1,2,3)        138: (4,2,2)
     32: (6)             53: (1,2,2,1)      139: (4,2,1,1)

Looking at length instead of parts gives A096199.
These composition are counted by A100346.
A version counting subsets instead of compositions is A125297.
An unordered version is A326841, counted by A018818.
A011782 counts compositions.
A316413 ranks partitions with sum divisible by length, counted by A067538.
A319333 ranks partitions with sum equal to lcm, counted by A074761.
Cf. A003242, A056239, A238279, A326836, A326842.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Permutations are ranked by A333218.
- Relatively prime compositions are ranked by A291166*, complement A291165.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],#==0||Divisible[Total[stc[#]],LCM@@stc[#]]&]

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

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

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

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A330225 Position of first appearance of n in A290103 = LCM of prime indices.

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A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

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A319330 Number of integer partitions of n whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts.

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A319334 Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.

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A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

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