A130714 -id:A130714 - cp's OEIS frontend

A083710 Number of integer partitions of n with a part dividing all the other parts.

1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 37, 43, 70, 78, 114, 143, 196, 232, 330, 386, 530, 641, 836, 1003, 1340, 1581, 2037, 2461, 3127, 3719, 4746, 5605, 7038, 8394, 10376, 12327, 15272, 17978, 22024, 26095, 31730, 37339, 45333, 53175, 64100, 75340, 90138

Offset: 0

N. J. A. Sloane, Jun 16 2003

Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part." - Joerg Arndt, Jun 08 2009
The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5) - 1 = 6, a(7) = A000041(7) - 3 = 12. - Vladeta Jovovic, Jun 17 2003
Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041. - Gary W. Adamson, Jun 08 2009

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (331)
                    (211)   (311)    (51)      (421)
                    (1111)  (2111)   (222)     (511)
                            (11111)  (321)     (2221)
                                     (411)     (3211)
                                     (2211)    (4111)
                                     (3111)    (22111)
                                     (21111)   (31111)
                                     (111111)  (211111)
                                               (1111111)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Cf. A018783, A137587.
Cf. A000041, A051731. - Gary W. Adamson, Jun 08 2009
The case with no 1's is A083711.
The strict case is A097986.
The version for "divisible by" instead of "dividing" is A130689.
The case where there is also a part divisible by all the others is A130714.
The complement of these partitions is counted by A338470.
The Heinz numbers of these partitions are dense, complement of A342193.
The case where there is also no part divisible by all the others is A343345.
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).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A001792, A098965, A264401, A339563, A343340, A343341, A343378.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-0 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11, ...). - Gary W. Adamson, Jan 27 2008
G.f.: 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n. - Joerg Arndt, Jun 08 2009
Gary W. Adamson's comment is equivalent to the formula a(n) = Sum_{d|n} p(d-1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d-1)*x^d/(1-x^d), - N. J. A. Sloane, Jun 08 2009

More terms from Vladeta Jovovic, Jun 17 2003

Name shortened by Gus Wiseman, Apr 18 2021

A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 28, 41, 43, 56, 65, 82, 88, 115, 122, 155, 174, 209, 225, 283, 305, 363, 402, 477, 514, 622, 666, 783, 858, 990, 1078, 1268, 1362, 1561, 1708, 1958, 2111, 2433, 2613, 2976, 3247, 3652, 3938, 4482, 4821, 5422

Offset: 0

Vladeta Jovovic, Jul 01 2007

First differs from A130714 at a(11) = 28, A130714(11) = 27. - Gus Wiseman, Apr 23 2021

			For n = 6 we have 10 such partitions: [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 2, 2], [2, 2, 2], [1, 1, 1, 3], [3, 3], [1, 1, 4], [2, 4], [1, 5], [6].
From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (62)
                    (211)   (311)    (51)      (421)      (71)
                    (1111)  (2111)   (222)     (511)      (422)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (4111)     (2222)
                                     (3111)    (22111)    (3311)
                                     (21111)   (31111)    (4211)
                                     (111111)  (211111)   (5111)
                                               (1111111)  (22211)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from Andrew Howroyd)

Cf. A018818, A117086.
The dual version is A083710.
The case without 1's is A339619.
The Heinz numbers of these partitions are the complement of A343337.
The complement is counted by A343341.
The strict case is A343347.
The complement in the strict case is counted by A343377.
A000009 counts strict partitions.
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.
A072233 counts partitions by sum and greatest part.
Cf. A066186, A083711, A097986, A338470, A341450, A343346, A343382.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

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

G.f.: 1 + Sum_{n>0} x^n/Product_{d divides n} (1-x^d).

A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 74, 78, 79, 82, 83, 86, 87, 89, 94, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113, 114, 115, 118, 122, 127

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

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 strict integer partitions whose smallest part divides all the others (counted by A097986). 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:
      2: {1}       29: {10}        59: {17}
      3: {2}       30: {1,2,3}     61: {18}
      5: {3}       31: {11}        62: {1,11}
      6: {1,2}     34: {1,7}       65: {3,6}
      7: {4}       37: {12}        66: {1,2,5}
     10: {1,3}     38: {1,8}       67: {19}
     11: {5}       39: {2,6}       70: {1,3,4}
     13: {6}       41: {13}        71: {20}
     14: {1,4}     42: {1,2,4}     73: {21}
     17: {7}       43: {14}        74: {1,12}
     19: {8}       46: {1,9}       78: {1,2,6}
     21: {2,4}     47: {15}        79: {22}
     22: {1,5}     53: {16}        82: {1,13}
     23: {9}       57: {2,8}       83: {23}
     26: {1,6}     58: {1,10}      86: {1,14}

These partitions are counted by A097986 (non-strict: A083710).
The case with no 1's is counted by A098965 (non-strict: A083711).
The squarefree complement is A339562, ranked by A341450.
The complement of the not necessarily squarefree version is A342193.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A005117 lists squarefree numbers.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices, row sums of A112798.
A338470 counts partitions with no dividing part.
Cf. A130714, A253249, A257993, A299702, A339564, A343340, A343378.

Select[Range[2,100],SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]},And@@IntegerQ/@(p/Min@@p)]&]

A083711 a(n) = A083710(n) - A000041(n-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 14, 1, 13, 8, 20, 1, 33, 1, 40, 14, 44, 1, 85, 6, 79, 25, 117, 1, 181, 1, 196, 45, 233, 17, 389, 1, 387, 80, 545, 1, 750, 1, 839, 165, 1004, 1, 1516, 12, 1612, 234, 2040, 1, 2766, 48, 3142, 388, 3720, 1, 5295, 1, 5606, 663, 7038, 83, 9194, 1, 10379, 1005

Offset: 1

N. J. A. Sloane, Jun 16 2003

Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(6) = 4 through a(12) = 13 partitions:
  (6)      (7)  (8)        (9)      (10)         (11)  (12)
  (3,3)         (4,4)      (6,3)    (5,5)              (6,6)
  (4,2)         (6,2)      (3,3,3)  (8,2)              (8,4)
  (2,2,2)       (4,2,2)             (4,4,2)            (9,3)
                (2,2,2,2)           (6,2,2)            (10,2)
                                    (4,2,2,2)          (4,4,4)
                                    (2,2,2,2,2)        (6,3,3)
                                                       (6,4,2)
                                                       (8,2,2)
                                                       (3,3,3,3)
                                                       (4,4,2,2)
                                                       (6,2,2,2)
                                                       (4,2,2,2,2)
                                                       (2,2,2,2,2,2)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

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

Allowing 1's gives A083710.
The strict case is A098965.
The complement (except also without 1's) is counted by A338470.
The dual version is A339619.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
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. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od: # James Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
seq(a(n), n=1..69);  # Alois P. Heinz, Feb 15 2023

a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 15 2023 *)

a(n) = Sum_{ d|n, dA000041(d-1).

More terms from James Sellers, Jun 21 2003

A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 12, 7, 22, 20, 32, 34, 60, 54, 98, 93, 145, 159, 237, 229, 361, 384, 529, 574, 810, 840, 1194, 1275, 1703, 1886, 2484, 2660, 3566, 3909, 4987, 5520, 7092, 7737, 9907, 10917, 13603, 15226, 18910, 20801, 25912, 28797

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of integer partitions of n that are either empty or have smallest part not dividing all the others and greatest part not divisible by all the others.

			The a(0) = 1 through a(12) = 7 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
                           (52)   (332)  (72)    (73)    (74)     (543)
                           (322)         (432)   (433)   (83)     (552)
                                         (522)   (532)   (92)     (732)
                                         (3222)  (3322)  (443)    (4332)
                                                         (533)    (5322)
                                                         (542)    (33222)
                                                         (722)
                                                         (3332)
                                                         (4322)
                                                         (5222)
                                                         (32222)

The opposite version is A130714.
The first condition alone gives A338470.
The Heinz numbers of these partitions are A343338 = A342193 /\ A343337.
The second condition alone gives A343341.
The half-opposite versions are A343344 and A343345.
The "or" instead of "and" version is A343346 (strict: A343382).
The strict case is A343379.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part (strict: A015723).
Cf. A006128, A066186, A083710, A130689, A341450, A343343, A343377.

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

A343346 Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 29, 36, 59, 80, 112, 150, 214, 271, 374, 476, 624, 800, 1045, 1298, 1669, 2088, 2628, 3258, 4087, 5000, 6219, 7602, 9331, 11368, 13877, 16754, 20368, 24536, 29580, 35468, 42624, 50845, 60827, 72357, 86078, 102100, 121101

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

First differs from A343345 at a(14) = 80, A343345(14) = 79.

Alternative name: Number of integer partitions of n with either no part dividing, or no part divisible by all the others.

			The a(0) = 1 through a(10) = 16 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (32)  (321)  (43)    (53)     (54)      (64)
                               (52)    (332)    (72)      (73)
                               (322)   (431)    (432)     (433)
                               (3211)  (521)    (522)     (532)
                                       (3221)   (531)     (541)
                                       (32111)  (3222)    (721)
                                                (3321)    (3322)
                                                (4311)    (4321)
                                                (5211)    (5221)
                                                (32211)   (5311)
                                                (321111)  (32221)
                                                          (33211)
                                                          (43111)
                                                          (52111)
                                                          (322111)
                                                          (3211111)

The complement is counted by A130714.
The first condition alone gives A338470.
The second condition alone gives A343341.
The "and" instead of "or" version is A343342.
The Heinz numbers of these partitions are A343343.
The strict case is A343382.
A000009 counts strict partitions.
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.
Cf. A083710, A130689, A341450, A342193, A343337, A343338, A343379.

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

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

Original entry on oeis.org

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

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}]

A343345 Number of integer partitions of n that are empty, or have smallest part dividing all the others, but do not have greatest part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 29, 36, 59, 79, 115, 149, 216, 270, 379, 473, 634, 793, 1063, 1292, 1689, 2079, 2667, 3241, 4142, 4982, 6291, 7582, 9434, 11321, 14049, 16709, 20545, 24490, 29860, 35380, 43004, 50741, 61282, 72284, 86680, 101906, 121990

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

First differs from A343346 at a(14) = 79, A343346(14) = 80.

Alternative name: Number of integer partitions of n with a part dividing all the others, but with no part divisible by all the others.

			The a(6) = 1 through a(11) = 16 partitions:
  (321)  (3211)  (431)    (531)     (541)      (641)
                 (521)    (3321)    (721)      (731)
                 (3221)   (4311)    (4321)     (4331)
                 (32111)  (5211)    (5221)     (5321)
                          (32211)   (5311)     (5411)
                          (321111)  (32221)    (7211)
                                    (33211)    (33221)
                                    (43111)    (43211)
                                    (52111)    (52211)
                                    (322111)   (53111)
                                    (3211111)  (322211)
                                               (332111)
                                               (431111)
                                               (521111)
                                               (3221111)
                                               (32111111)

The first condition alone gives A083710.
The half-opposite versions are A130714 and A343342.
The Heinz numbers of these partitions are 1 and A343340.
The second condition alone gives A343341.
The opposite version is A343344.
The strict case is A343381.
A000009 counts strict partitions.
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).
Cf. A083711, A097986, A098743, A098965, A130689, A264401, A339563, A343337.

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

A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 5, 1, 6, 4, 6, 7, 15, 6, 16, 15, 20, 17, 36, 18, 43, 36, 46, 48, 72, 45, 93, 82, 103, 88, 152, 104, 179, 158, 191, 194, 285, 202, 328, 292, 373, 348, 502, 391, 576, 519, 659, 634, 864, 665

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of integer partitions of n with no part dividing all the others, but with a part divisible by all the others.

			The a(18) = 1  through a(23) = 15 partitions (A..E = 10..14):
  633222   C43       C332      C432       C64        E72
           A522      66332     A5222      A552       F53
           C322      633332    C3222      C433       I32
           66322     6332222   663222     C3322      C443
           633322              6333222    663322     C632
           6322222             63222222   6333322    66632
                                          63322222   C3332
                                                     C4322
                                                     663332
                                                     A52222
                                                     C32222
                                                     6333332
                                                     6632222
                                                     63332222
                                                     632222222

The second condition alone gives A130689.
The half-opposite versions are A130714 and A343342.
The first condition alone gives A338470.
The Heinz numbers of these partitions are 1 and A343339.
The opposite version is A343345.
The strict case is A343380.
A000009 counts strict partitions.
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.
Cf. A083710, A097986, A264401, A339562, A341450, A342193, A343346.

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

cp's OEIS Frontend

A083710 Number of integer partitions of n with a part dividing all the other parts.

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A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1.

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A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.

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A083711 a(n) = A083710(n) - A000041(n-1).

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A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

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A343346 Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.

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

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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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