A343381 -id:A343381 - cp's OEIS frontend

A097986 Number of strict integer partitions of n with a part dividing all the other parts.

1, 1, 2, 2, 2, 4, 3, 5, 5, 7, 6, 12, 9, 13, 15, 20, 18, 28, 26, 37, 39, 47, 49, 71, 68, 85, 94, 117, 120, 159, 160, 201, 216, 257, 277, 348, 357, 430, 470, 562, 592, 720, 758, 901, 981, 1134, 1220, 1457, 1542, 1798, 1952, 2250, 2419, 2819, 3023, 3482, 3773, 4291

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 23 2021

Also the number of uniform (constant multiplicity) partitions of n containing 1, ranked by A367586. The strict case is A096765. The version without 1 is A329436. - Gus Wiseman, Dec 01 2023

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (2,1)  (3,1)  (4,1)  (4,2)    (6,1)    (6,2)
                                 (5,1)    (4,2,1)  (7,1)
                                 (3,2,1)           (4,3,1)
                                                   (5,2,1)
The a(1) = 1 through a(8) = 5 uniform partitions containing 1:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (11111)  (321)     (421)      (431)
                                     (2211)    (1111111)  (521)
                                     (111111)             (3311)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 500 terms from John Tyler Rascoe)

The non-strict version is A083710.
The case with no 1's is A098965.
The Heinz numbers of these partitions are A339563.
The strict complement is counted by A341450.
The version for "divisible by" instead of "dividing" is A343347.
The case where there is also a part divisible by all the others is A343378.
The case where there is no part divisible by all the others is A343381.
A000005 counts divisors.
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.
Cf. A083711, A098743, A130689, A200745, A264401, A338470, A343377, A343379, A343380.
Cf. A023645, A025147, A047966, A072774, A096765, A329436, A367586.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x,#}]&]], {n,0,30}] (* Gus Wiseman, Apr 23 2021 *)

A_x(N) = {my(x='x+O('x^N)); Vec(sum(k=1,N,x^k*prod(i=2,N-k, (1+x^(k*i)))))}
A_x(50) \\ John Tyler Rascoe, Nov 19 2024

a(n) = Sum_{d|n} A025147(d-1).
G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 06 2025

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

Name shortened by Gus Wiseman, Apr 23 2021

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 3, 9, 9, 12, 12, 20, 18, 28, 27, 37, 42, 55, 51, 74, 80, 98, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of strict integer partitions of n with no part dividing all the others.

			The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
  0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
                         52        72    73    74    543   85    95     96
                                   432   532   83    732   94    A4     B4
                                               92          A3    B3     D2
                                               542         B2    653    654
                                               632         643   743    753
                                                           652   752    762
                                                           742   932    843
                                                           832   5432   852
                                                                        942
                                                                        A32
                                                                        6432

The complement is counted by A097986 (non-strict: A083710, rank: A339563).
The complement with no 1's is A098965 (non-strict: A083711).
The non-strict version is A338470.
The Heinz numbers of these partitions are A339562 (non-strict: A342193).
The case with greatest part not divisible by all others is A343379.
The case with greatest part divisible by all others is A343380.
A000009 counts strict partitions (non-strict: A000041).
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

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

a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).

A343341 Number of integer partitions of n with no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 28, 36, 58, 79, 111, 149, 209, 270, 368, 472, 618, 793, 1030, 1292, 1653, 2073, 2608, 3241, 4051, 4982, 6176, 7566, 9285, 11320, 13805, 16709, 20275, 24454, 29477, 35380, 42472, 50741, 60648, 72199, 85887, 101906, 120816

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

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

			The a(5) = 1 through a(10) = 16 partitions:
  (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 A130689.
The dual version is A338470.
The Heinz numbers of these partitions are A343337.
The strict case is 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.
Cf. A066186, A083710, A083711, A097986, A098965, A341450, A343342, A343345, A343346, A343381, A343382.

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

A343337 Numbers with 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, 198, 201, 203, 204, 205, 207, 209, 210, 215

Offset: 1

Gus Wiseman, Apr 13 2021

nonn

Alternative name: 1 and numbers whose greatest prime index is not divisible by all the other prime indices.
First differs from A318992 in lacking 195.
First differs from A343343 in lacking 195.
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 with greatest part not divisible by all the others (counted by A343341). 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, 195 has prime indices {2,3,6}, and 6 is divisible by both 2 and 3, so 195 does not belong to the sequence.

The complement is counted by A130689.
The dual version is A342193.
The case with smallest prime index not dividing all the others is A343338.
The case with smallest prime index dividing by all the others is A343340.
These are the Heinz numbers of the partitions counted by A343341.
Including the dual version gives A343343.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
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, A341450, A343342, A343345, A343346, A343377, A343379, A343381, A343382.

Select[Range[1000],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)]&]

A343377 Number of strict integer partitions of n with no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 9, 13, 18, 21, 26, 32, 38, 47, 57, 66, 80, 95, 110, 132, 157, 181, 211, 246, 282, 327, 379, 435, 500, 570, 648, 743, 849, 963, 1094, 1241, 1404, 1592, 1799, 2025, 2282, 2568, 2882, 3239, 3634, 4066, 4554, 5094, 5686, 6346

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

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

			The a(5) = 1 through a(12) = 9 partitions:
  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)      (7,5)
                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)      (5,4,3)
                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)      (6,4,2)
                                  (5,3,1)  (5,4,1)    (9,2)      (6,5,1)
                                           (7,2,1)    (5,4,2)    (7,3,2)
                                           (4,3,2,1)  (6,4,1)    (7,4,1)
                                                      (7,3,1)    (8,3,1)
                                                      (5,3,2,1)  (9,2,1)
                                                                 (5,4,2,1)

The dual strict complement is A097986.
The dual version is A341450.
The non-strict version is A343341 (Heinz numbers: A343337).
The strict complement is counted by A343347.
The case with smallest part not divisible by all the others is A343379.
The case with smallest part divisible by all the others is A343381.
A000005 counts divisors.
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, A200745, A264401, A338470, A339562, A343338, A343342, A343345, A343346, A343382.

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

A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 3, 9, 9, 12, 12, 18, 18, 27, 27, 36, 41, 51, 51, 73, 80, 96, 105, 132, 137, 177, 188, 230, 253, 303, 320, 398, 431, 508, 550, 659, 705, 847, 913, 1063, 1165, 1359, 1452, 1716, 1856, 2134, 2329, 2688, 2894, 3345, 3622, 4133

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.

			The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
  (3,2)  .  (4,3)  (5,3)  (5,4)    (6,4)    (6,5)    (7,5)    (7,6)
            (5,2)         (7,2)    (7,3)    (7,4)    (5,4,3)  (8,5)
                          (4,3,2)  (5,3,2)  (8,3)    (7,3,2)  (9,4)
                                            (9,2)             (10,3)
                                            (5,4,2)           (11,2)
                                                              (6,4,3)
                                                              (6,5,2)
                                                              (7,4,2)
                                                              (8,3,2)

The first condition alone gives A341450.
The non-strict version is A343342  (Heinz numbers: A343338).
The second condition alone gives A343377.
The opposite version is A343378.
The half-opposite versions are A343380 and A343381.
The version for "or" instead of "and" is 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, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.

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

The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.

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}]

A343380 Number of strict integer partitions of n with no part dividing all the others but with a part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 4, 0, 1, 0, 2, 0, 4, 0, 3, 1, 2, 2, 5, 0, 5, 3, 4, 1, 9, 1, 5, 2, 4, 5, 11, 1, 6, 4, 11, 3, 13, 5, 10, 4, 11, 8, 14, 3, 10, 6, 9, 3, 15, 6, 14, 10, 18, 8

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either empty or (1) have smallest part not dividing all the others and (2) have greatest part divisible by all the others.

			The a(11) = 1 through a(29) = 4 partitions (empty columns indicated by dots, A..O = 10..24):
  632  .  .  .  .  .  A52  .  C43  .  C432  C64  E72   .  C643  .  K52    .  I92
                      C32                        F53               C6432     K54
                                                 I32                         O32
                                                 C632                        I632

The first condition alone gives A341450.
The non-strict version is A343344 (Heinz numbers: A343339).
The second condition alone gives A343347.
The half-opposite versions are A343378 and A343379.
The opposite (and dual) version 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).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343377, A343382.

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

cp's OEIS Frontend

A097986 Number of strict integer partitions of n with a part dividing all the other parts.

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A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

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A343341 Number of integer partitions of n with no part divisible by all the others.

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A343337 Numbers with no prime index divisible by all the other prime indices.

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A343377 Number of strict integer partitions of n with no part divisible by all the others.

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A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

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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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