A018818 -id:A018818 - cp's OEIS frontend

A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 5, 1, 7, 1, 8, 4, 6, 1, 15, 2, 9, 5, 14, 1, 22, 1, 20, 7, 18, 4, 36, 1, 26, 10, 40, 1, 51, 1, 48, 18, 49, 1, 86, 3, 73, 19, 86, 1, 117, 7, 120, 27, 120, 1, 196, 1, 160, 42, 201, 10, 259, 1, 258, 50, 292, 1, 407, 1, 357, 81, 431, 8, 548, 1, 577

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume this part is the smallest. - Gus Wiseman, Apr 18 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

The non-strict version with 1's allowed is A083710.
The non-strict version is A083711.
The version with 1's allowed is A097986.
The Heinz numbers of these partitions are the odd terms of A339563.
The non-strict dual is A339619.
The strict complement is counted by A341450.
A000005 counts divisors.
A000041 counts partitions.
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).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 92}], {k, 2, 92}]], x], {2, 81}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

a(n) = Sum_{d|n, dA025147(d-1).

G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A326851 Number of strict integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 5, 1, 6, 1, 1, 1, 16, 1, 1, 1, 12, 1, 33, 1, 15, 1, 1, 1, 60, 1, 1, 1, 51, 1, 81, 1, 31, 57, 1, 1, 216, 1, 55, 1, 45, 1, 230, 1, 223, 1, 1, 1, 800, 1, 1, 314, 273, 1, 607, 1, 81, 1, 315, 1, 2404, 1, 1, 319

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

			The a(6) = 2 through a(24) = 16 partitions (1 terms not shown):
  6       12        15          16        18      20           24
  3,2,1   6,4,2     5,4,3,2,1   8,4,3,1   9,5,4   10,5,3,2     12,7,5
          6,5,1                 8,5,2,1   9,6,3   10,5,4,1     12,8,4
          6,3,2,1                         9,7,2   10,6,3,1     12,9,3
                                          9,8,1   10,7,2,1     12,10,2
                                                  10,4,3,2,1   12,11,1
                                                               8,7,5,4
                                                               8,7,6,3
                                                               12,5,4,3
                                                               12,6,4,2
                                                               12,6,5,1
                                                               12,7,3,2
                                                               12,7,4,1
                                                               12,8,3,1
                                                               12,9,2,1
                                                               8,6,4,3,2,1

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..383

The non-strict case is A326843.

Cf. A018818, A047993, A067538, A326837, A326842, A326849, A326852.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[n,Max[#]]&&Divisible[n,Length[#]]&]]],{n,0,30}]

A343347 Number of strict integer partitions of n with a part divisible by all the others.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   632  93    841  D1   C3
                                 621  631  821  A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The dual version is A097986 (non-strict: A083710).
The non-strict version is A130689 (Heinz numbers: complement of A343337).
The strict complement is counted by A343377.
The case with smallest part divisible by all the others is A343378.
The case with smallest part not divisible by all the others is A343380.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

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

seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021

A066882 Number of partitions of n into prime divisors of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 4, 1, 3, 2, 2, 1, 5, 1, 2, 1, 3, 1, 21, 1, 1, 2, 2, 2, 7, 1, 2, 2, 5, 1, 28, 1, 3, 4, 2, 1, 9, 1, 6, 2, 3, 1, 10, 2, 5, 2, 2, 1, 71, 1, 2, 4, 1, 2, 42, 1, 3, 2, 43, 1, 13, 1, 2, 6, 3, 2, 49, 1, 9, 1, 2, 1, 97, 2, 2, 2, 5, 1, 151, 2, 3, 2, 2, 2, 17, 1, 8

Offset: 0

Naohiro Nomoto, Jan 26 2002

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

Cf. A018818, A014652.

Main diagonal of A107329 (for n>=1).

with(numtheory):
a:= proc(n) local b, l; l:= sort([factorset(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, Feb 05 2014

a[0] = 1; a[n_] := SeriesCoefficient[1/Product[1-x^d, {d, FactorInteger[n][[All, 1]]}], {x, 0, n}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 30 2015, after Vladeta Jovovic *)

from sympy import factorint
from functools import cache
def A066882(n):
    @cache
    def b(m, i):
        if m == 0: return 1
        if i < 0: return 0
        return b(m, i-1) + (0 if l[i]>m else b(m-l[i], i))
    l = sorted(factorint(n))
    return b(n, len(l)-1)
print([A066882(n) for n in range(99)]) # Michael S. Branicky, Jan 08 2025 after Alois P. Heinz

Coefficient of x^n in expansion of 1/Product_{d is prime divisor of n} (1-x^d). - Vladeta Jovovic, Apr 11 2004

More terms from Sascha Kurz, Mar 23 2002

Corrected by Vladeta Jovovic, Apr 11 2004

A326850 Number of strict integer partitions of n whose maximum part divides n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 10, 1, 10, 5, 12, 1, 23, 1, 18, 15, 23, 1, 49, 1, 34, 36, 38, 1, 106, 1, 54, 79, 81, 1, 189, 1, 124, 162, 104, 1, 412, 1, 145, 307, 289, 1, 608, 12, 437, 559, 256, 1, 1432, 1, 340, 981, 976, 79, 1730, 1

Offset: 0

Gus Wiseman, Jul 28 2019

nonn

			The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5)
   6: (6)
   6: (3,2,1)
   7: (7)
   8: (8)
   8: (4,3,1)
   9: (9)
  10: (10)
  10: (5,4,1)
  10: (5,3,2)
  11: (11)
  12: (12)
  12: (6,5,1)
  12: (6,4,2)
  12: (6,3,2,1)
  13: (13)
  14: (14)
  14: (7,6,1)
  14: (7,5,2)
  14: (7,4,3)
  14: (7,4,2,1)
  15: (15)
  15: (5,4,3,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

Positions of 1's appear to be A308168.
The non-strict case is given by A067538.
Cf. A018818, A033630, A067538, A102627, A200745, A316413, A326625, A326836, A326843, A326851.

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

A339662 Greatest gap in the partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 3, 1, 0, 6, 0, 7, 2, 3, 4, 8, 0, 2, 5, 1, 3, 9, 0, 10, 0, 4, 6, 2, 0, 11, 7, 5, 2, 12, 3, 13, 4, 1, 8, 14, 0, 3, 2, 6, 5, 15, 0, 4, 3, 7, 9, 16, 0, 17, 10, 3, 0, 5, 4, 18, 6, 8, 2, 19, 0, 20, 11, 1, 7, 3, 5, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
FindStat, Dyson's crank of a partition.
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)

Positions of first appearances are A000040.
Positions of 0's are A055932.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The version for crank is A257989.
The minimal instead of maximal version is A257993.
The version for greatest difference is A286469 or A286470.
Positive integers by Heinz weight and image are counted by A339737.
Positions of 1's are A339886.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[maxgap[primeMS[n]],{n,100}]

a(n) = A000720(A079068(n)).

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

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

A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 6, 5, 4, 6, 6, 4, 8, 6, 7, 9, 8, 5, 12, 9, 8, 9, 11, 6, 14, 10, 10, 11, 10, 10, 20, 12, 12, 15, 18, 10, 21, 13, 15, 19, 17, 11, 27, 19, 20, 20, 25, 13, 27, 22, 26, 23, 24, 15, 34, 23, 21, 27, 30, 19, 38, 24, 26, 27, 37

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n with a part dividing all the others and a part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   821  93    841  D1   C3
                                 621  631       A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

The first condition alone gives A097986.
The non-strict version is A130714 (Heinz numbers are complement of A343343).
The second condition alone gives A343347.
The opposite version is A343379.
The half-opposite versions are A343380 and A343381.
The strict complement is counted by A343382.
A000009 counts strict partitions.
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).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A264401, A339562, A339563, A341450, A343346, A343377.

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

A225244 Number of partitions of n into squarefree divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 8, 2, 5, 4, 11, 2, 27, 2, 14, 14, 9, 2, 64, 2, 40, 18, 20, 2, 125, 6, 23, 10, 53, 2, 742, 2, 17, 26, 29, 26, 343, 2, 32, 30, 195, 2, 1654, 2, 79, 136, 38, 2, 729, 8, 341, 38, 92, 2, 1000, 38, 265, 42, 47, 2, 14188, 2, 50, 184, 33, 44, 5257, 2

Offset: 0

Reinhard Zumkeller, May 05 2013

nonn

a(n) <= A018818(n);
a(n) = A018818(n) iff n is squarefree: a(A005117(n)) = A018818(A005117(n));
a(A000040(n)) = 2.

			a(8) = #{2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+6x1, 8x1} = 5;
a(9) = #{3+3+3, 3+3+1+1+1, 3+1+1+1+1+1+1, 9x1} = 4;
a(10) = #{10, 5+5, 5+2+2+1, 5+2+1+1+1, 5+5x1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1, 2+2+6x1, 2+8x1, 10x1} = 11;
a(11) = #{11, 1+1+1+1+1+1+1+1+1+1+1} = 2;
a(12) = #{6+6, 6+3+3, 6+3+2+1, 6+3+1+1+1, 6+2+2+2, 6+2+2+1+1, 6+2+1+1+1+1, 6+6x1, 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1, 3+3+2+4x1, 3+3+6x1, 3+2+2+2+2+1, 3+2+2+2+1+1+1, 3+2+2+5x1, 3+2+7x1, 3+8x1, 2+2+2+2+2+2, 2+2+2+2+2+1+1, 2+2+2+2+1+1+1+1, 2+2+2+6x1, 2+2+8x1, 2+10x1, 12x1} = 27;
a(13) = #{11, 1+1+1+1+1+1+1+1+1+1+1+1+1} = 2;
a(14) = #{14, 7+7, 7+2+2+2+1, 7+2+2+1+1+1, 7+2+5x1, 7+7x1, 7x2, 6x2+1+1, 5x2+1+1+1+1, 4x2+6x1, 2+2+2+8x1, 2+2+10x1, 2+12x1, 14x1} = 14;
a(15) = #{15, 5+5+5, 5+5+3+1+1, 5+5+5x1, 5+3+3+3+1, 5+3+3+1+1+1+1, 5+3+7x1, 5+10x1, 3+3+3+3+3, 3+3+3+3+1+1+1, 3+3+3+6x1, 3+3+9x1, 3+12x1, 15x1} = 14.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (300 terms from Reinhard Zumkeller)

Cf. A206778, A005117, A008966, A225245, A073576.

a225244 n = p (a206778_row n) n where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

with(numtheory):
a:= proc(n) local b, l; l:= sort([select(issqrfree, divisors(n))[]]):
      b:= proc(m, i) option remember; `if`(m=0 or i=1, 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, Feb 05 2014

a[0] = 1; a[n_] := Module[{b, l}, l = Select[Divisors[n], SquareFreeQ]; b[m_, i_] := b[m, i] = If[m == 0 || i == 1, 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, Oct 27 2015, after Alois P. Heinz *)

a(n) = [x^n] Product_{d|n, mu(d) != 0} 1/(1 - x^d), where mu() is the Moebius function (A008683). - Ilya Gutkovskiy, Jul 26 2017

A279788 Twice partitioned numbers where the first partition is constant and the latter partitions are strict.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 10, 6, 12, 17, 21, 13, 57, 19, 49, 87, 86, 39, 240, 55, 279, 330, 235, 105, 1141, 386, 491, 1217, 1461, 257, 4804, 341, 2968, 4225, 1958, 5898, 18961, 761, 3782, 15007, 30572, 1261, 66245, 1611, 32523, 106951, 13122, 2591, 283013, 81390, 182873

Offset: 0

Gus Wiseman, Dec 18 2016

nonn

			The a(6)=10 twice-partitions are:
((6)), ((51)), ((42)), ((3)(3)), ((3)(21)), ((21)(3)),
((321)), ((2)(2)(2)), ((21)(21)), ((1)(1)(1)(1)(1)(1)).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf. A000005, A000009, A018818, A063834, A279787.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(n/d)^d, d=divisors(n)))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

Table[DivisorSum[n,PartitionsQ[n/#]^#&],{n,20}]

cp's OEIS Frontend

A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

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A326851 Number of strict integer partitions of n whose length and maximum both divide n.

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A343347 Number of strict integer partitions of n with a part divisible by all the others.

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A066882 Number of partitions of n into prime divisors of n.

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A326850 Number of strict integer partitions of n whose maximum part divides n.

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A339662 Greatest gap in the partition with Heinz number n.

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A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

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A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

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