A018818 -id:A018818 - cp's OEIS frontend

A284345 Number of partitions of n into squares dividing n.

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 6, 1, 3, 1, 6, 1, 1, 1, 7, 2, 1, 4, 8, 1, 1, 1, 15, 1, 1, 1, 27, 1, 1, 1, 11, 1, 1, 1, 12, 6, 1, 1, 28, 2, 3, 1, 14, 1, 7, 1, 15, 1, 1, 1, 16, 1, 1, 8, 46, 1, 1, 1, 18, 1, 1, 1, 114, 1, 1, 4, 20, 1, 1, 1, 66, 11, 1, 1, 22, 1, 1, 1, 23, 1, 11, 1, 24, 1, 1, 1, 91, 1, 3, 12, 67

Offset: 0

Ilya Gutkovskiy, Mar 25 2017

nonn

			a(8) = 3 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are squares {1, 4} therefore we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A001156, A018818, A046951, A066882, A161148, A225244, A284289.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:=
      sort(select(issqr, [divisors(n)[]])),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[Mod[DivisorSigma[0, d[[k]]], 2] == 1] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 100}]]

a(n) = [x^n] Product_{d^2|n} 1/(1 - x^(d^2)).
a(n) = 1 if n is a squarefree.
a(n) = 2 if n is a square of prime.

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

Original entry on oeis.org

1, 15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

After 1, first differs from A318992 in lacking 390, with prime indices {1,2,3,6}.
First differs from A343337 in having 195, with prime indices {2,3,6}.
Alternative name: 1 and numbers where either the smallest prime index is not a divisor of all the other prime indices, or the greatest prime index is not divisible by all the other prime indices.
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.
Also Heinz numbers of partitions that either empty, have smallest part not dividing all the others, or have greatest part not divisible by all the others (counted by A343346). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            90: {1,2,2,3}      141: {2,15}
     15: {2,3}         91: {4,6}          143: {5,6}
     30: {1,2,3}       93: {2,11}         145: {3,10}
     33: {2,5}         95: {3,8}          150: {1,2,3,3}
     35: {3,4}         99: {2,2,5}        153: {2,2,7}
     45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
     51: {2,7}        105: {2,3,4}        155: {3,11}
     55: {3,5}        110: {1,3,5}        161: {4,9}
     60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
     66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
     69: {2,9}        123: {2,13}         175: {3,3,4}
     70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
     75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
     77: {4,5}        138: {1,2,9}        182: {1,4,6}
     85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
For example, the prime indices of 90 are {1,2,2,3}, and, because 1 divides all the other parts, 90 is in the sequence, even though 3 is not divisible by all the other parts.

The partitions without these Heinz numbers are counted by A130714.
The first condition alone gives A342193.
The second condition alone gives A343337.
The "and" instead of "or" version is A343338.
The partitions with these Heinz numbers are counted by A343346.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A083710, A257993, A338470, A339562, A341450, A343339, A343340, A343341, A343342, A343378, A343379, A343382.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)||!And@@IntegerQ/@(p/Min@@p)]&]

Equals the union of A342193 and A343337.

A171565 Number of partitions of n into odd divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 1, 5, 3, 2, 5, 2, 3, 14, 1, 2, 12, 2, 5, 18, 3, 2, 9, 7, 3, 23, 5, 2, 54, 2, 1, 26, 3, 26, 35, 2, 3, 30, 9, 2, 72, 2, 5, 286, 3, 2, 17, 9, 18, 38, 5, 2, 93, 38, 9, 42, 3, 2, 275, 2, 3, 493, 1, 44, 108, 2, 5, 50, 110, 2, 117, 2, 3, 698, 5, 50, 126, 2, 17, 239, 3, 2, 375, 56

Offset: 0

Reinhard Zumkeller, Dec 11 2009

nonn

a(2*n+1) = A018818(2*n+1), a(A005408(n))=A018818(A005408(n));
a(2^k) = 1, a(A000079(n))=1;
for odd primes p: a(p*2^k) = 2^k + 1,
especially for n>1: a(A000040(n))=2, a(A100484(n))=3, a(A001749(n))=5.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A038550, A065091.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> is(x:: odd), divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

a[0] = 1; a[n_] := a[n] = Module[{b, l}, l = Select[Divisors[n], OddQ]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i < 1, 0, b[m, i-1] + If[l[[i]] > m, 0, b[m - l[[i]], i]]]]; b[n, Length[l]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)

a(n) = f(n,n,1) with f(n,m,k) = if k<=m then f(n,m,k+2)+f(n,m-k,k)*0^(n mod k) else 0^m.

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

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

A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892

Offset: 0

Gus Wiseman, Jan 23 2021

nonn

The only strict partitions counted are (), (1), and (2,1).

Is there a simple generating function?

			The a(1) = 1 through a(9) = 7 partitions:
  1  11  21   22    311    2211    331      2222      333
         111  1111  2111   111111  2221     4211      4221
                    11111          4111     221111    51111
                                   211111   311111    222111
                                   1111111  11111111  321111
                                                      21111111
                                                      111111111

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428), with strict case A340830.
The case where length also divides n is A326842 (A326847).
The Heinz numbers of these partitions are A340606.
The version for factorizations is A340851, with reciprocal version A340853.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A067539 counts partitions with integer geometric mean (A326623).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A330950 = partitions of n whose Heinz number is divisible by n (A324851).
Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]

A293813 Number of partitions of n into nontrivial divisors of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 11, 0, 2, 2, 9, 0, 14, 0, 15, 2, 2, 0, 79, 1, 2, 4, 19, 0, 93, 0, 35, 2, 2, 2, 279, 0, 2, 2, 157, 0, 153, 0, 27, 24, 2, 0, 1075, 1, 28, 2, 31, 0, 254, 2, 261, 2, 2, 0, 7025, 0, 2, 31, 201, 2, 320, 0, 39, 2, 301, 0, 12071, 0, 2, 35, 43, 2, 427, 0, 3073

Offset: 0

Ilya Gutkovskiy, Oct 16 2017

nonn

			a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].

Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the b-file of A211110 provided by Alois P. Heinz)
Index entries for sequences related to partitions

Cf. A018818, A027750, A070824, A210442, A211110, A293814.

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 16 2017

Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 80}]

a(n) = [x^n] Product_{d|n, 1 < d < n} 1/(1 - x^d).

a(n) = A211110(n) - 1 for n > 1.

A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 17, 1, 1, 1, 7, 1, 60, 1, 1, 1, 1, 1, 76, 1, 1, 1, 55, 1, 105, 1, 11, 10, 1, 1, 187, 1, 6, 1, 13, 1, 30, 1, 111, 1, 1, 1, 5043, 1, 1, 15, 1, 1, 230, 1, 17, 1, 242, 1, 4173, 1, 1, 12, 19, 1

Offset: 0

Gus Wiseman, Aug 01 2018

nonn

			The a(20) = 5 partitions are (20), (10,4,4,2), (10,4,2,2,2), (5,5,4,4,2), (5,5,4,2,2,2).
The a(45) = 10 partitions:
  (45),
  (15,15,9,3,3), (15,9,9,9,3),
  (15,9,9,3,3,3,3), (15,9,5,5,5,3,3), (9,9,9,5,5,5,3),
  (15,9,3,3,3,3,3,3,3), (9,9,5,5,5,3,3,3,3), (9,5,5,5,5,5,5,3,3),
  (9,5,5,5,3,3,3,3,3,3,3).
From _David A. Corneth_, Sep 08 2018: (Start)
Let sum(t) denote the sum of elements of a tuple t. The tuples t with distinct divisors of 45 that have lcm(t) = 45 and sum(t) <= 45 are {(45) and (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15), (5, 9, 15)}. For each such tuple t, find the number of partitions of 45 - s(t) into distinct parts of t.
For the tuple (45), there is 1 partition of 45 - 45 = 0 into parts with 45. That is: {()}.
For the tuple (3, 9, 15), there are 4 partitions of 45 - (3 + 9 + 15) = 18 into parts with 3, 9 and 15. They are {(3, 15), (9, 9), (3, 3, 3, 9), (3, 3, 3, 3, 3, 3)}.
For the tuple (3, 5, 9), there are 4 partitions of 45 - (3 + 5 + 9) = 28 into parts with 3, 5 and 9; they are {(5, 5, 9, 9), (3, 3, 3, 5, 5, 9), (3, 5, 5, 5, 5, 5), (3, 3, 3, 3, 3, 3, 5, 5)}.
For the tuple (3, 5, 9, 15), there is 1 partition of 45 - (3 + 5 + 9 + 15) = 13 into parts with 3, 5, 9 and 15. That is (3, 5, 5).
The other tuples, (5, 9), (9, 15), and (5, 9, 15); they give no extra tuples. That's because there is no solution to the Diophantine equation for 5x + 9y = 45 - (5 + 9), corresponding to the tuple (5, 9) with nonnegative x, y.
That also excludes (9, 15); if there is a solution for that, there would also be a solution for (5, 9). This could whittle down the number of seeds even further. Similarly, (5, 9, 15) gives no solution.
Therefore a(45) = 1 + 4 + 4 + 1 = 10.
(End)
In general, there are A318670(n) (<= A069626(n)) such seed sets of divisors where to start extending the partition from. (See the second PARI program which uses subroutine toplevel_starting_sets.) - _Antti Karttunen_, Sep 08 2018

Antti Karttunen, Table of n, a(n) for n = 0..359
Index entries for sequences related to lcm's

Cf. A000837, A018818, A066874, A067538, A069626, A074761, A074971, A143773, A259936, A281116, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A318670.

Table[Length[Select[IntegerPartitions[n],And[Min@@#>=2,LCM@@#==n]&]],{n,30}]

strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into_lcm(orgn,n,parts,from=1,m=1) = if(!n,(m==orgn),my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into_lcm(orgn,n-parts[i],parts,i,lcm(m,parts[i])))); (s));
A317624(n) = if(n<=1,0,partitions_into_lcm(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 07 2018

strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into(n,parts,from=1) = if(!n,1, if(#parts==from, (0==(n%parts[from])), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s)));
toplevel_starting_sets(orgn,n,parts,from=1,ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn,s += partitions_into(n,ss)); for(i=from,k,if(parts[i]<=n, newss = List(ss); listput(newss,parts[i]); s += toplevel_starting_sets(orgn,n-parts[i],parts,i+1,newss))); (s) };
A317624(n) = if(n<=1,0,toplevel_starting_sets(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 08-10 2018

A326852 Number of non-constant integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 16, 0, 1, 7, 11, 0, 30, 0, 40, 18, 1, 0, 201, 0, 1, 38, 124, 0, 426, 0, 211, 73, 1, 48, 1391, 0, 1, 131, 1741, 0, 1774, 0, 951, 2145, 1, 0, 8345, 0, 1853, 381, 2382, 0, 6718, 2761, 10633, 623, 1, 0, 68037

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

The Heinz numbers of these partitions are given by A326838.

			The a(6) = 1 through a(16) = 11 partitions (empty columns not shown):
  (321)  (4211)  (52111)  (633)     (7211111)  (53322)  (8332)
                          (642)                (53331)  (8422)
                          (651)                (54222)  (8431)
                          (4332)               (54321)  (8521)
                          (4422)               (54411)  (8611)
                          (4431)               (55221)  (42222211)
                          (6222)               (55311)  (43222111)
                          (6321)                        (43321111)
                          (6411)                        (44221111)
                          (322221)                      (44311111)
                          (332211)                      (82111111)
                          (333111)
                          (422211)
                          (432111)
                          (441111)
                          (621111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..180

The possibly constant case is A326843.
The strict case is A326851.
Cf. A018818, A047993, A067538, A326837, A326838, A326842, A326849.

Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&Divisible[n,Length[#]]&&Divisible[n,Max[#]]&]],{n,0,30}]

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

Gus Wiseman, Sep 25 2019

nonn

			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)

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.

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

cp's OEIS Frontend

A284345 Number of partitions of n into squares dividing n.

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A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

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A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

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A171565 Number of partitions of n into odd divisors of n.

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

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A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

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A293813 Number of partitions of n into nontrivial divisors of n.

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A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

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