A343338 -id:A343338 - cp's OEIS frontend

A342193 Numbers with no prime index dividing all the other prime indices.

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299

Offset: 1

Gus Wiseman, Apr 11 2021

nonn

Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.
First differs from A339562 in having 45.
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 1 and Heinz numbers of integer partitions with smallest part not dividing all the others (counted by A338470). 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: {}         105: {2,3,4}      201: {2,19}
     15: {2,3}      119: {4,7}        203: {4,10}
     33: {2,5}      123: {2,13}       205: {3,13}
     35: {3,4}      135: {2,2,2,3}    207: {2,2,9}
     45: {2,2,3}    141: {2,15}       209: {5,8}
     51: {2,7}      143: {5,6}        215: {3,14}
     55: {3,5}      145: {3,10}       217: {4,11}
     69: {2,9}      153: {2,2,7}      219: {2,21}
     75: {2,3,3}    155: {3,11}       221: {6,7}
     77: {4,5}      161: {4,9}        225: {2,2,3,3}
     85: {3,7}      165: {2,3,5}      231: {2,4,5}
     91: {4,6}      175: {3,3,4}      245: {3,4,4}
     93: {2,11}     177: {2,17}       247: {6,8}
     95: {3,8}      187: {5,7}        249: {2,23}
     99: {2,2,5}    195: {2,3,6}      253: {5,9}

The complement is counted by A083710 (strict: A097986).
The complement with no 1's is A083711 (strict: A098965).
These partitions are counted by A338470 (strict: A341450).
The squarefree case is A339562, with squarefree complement A339563.
The case with maximum prime index not divisible by all others is A343338.
The case with maximum prime index divisible by all others is A343339.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices, row sums of A112798.
A299702 lists Heinz numbers of knapsack partitions.
A339564 counts factorizations with a selected factor.
Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.

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

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)]&]

A343382 Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 9, 13, 18, 21, 26, 34, 38, 48, 57, 67, 81, 99, 110, 133, 157, 183, 211, 250, 282, 330, 380, 437, 502, 575, 648, 748, 852, 967, 1095, 1250, 1405, 1597, 1801, 2029, 2287, 2579, 2883, 3245, 3638, 4077, 4557, 5107, 5691, 6356

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either (1) empty, or (2) have smallest part not dividing all the others, or (3) have greatest part not divisible by all the others.

			The a(0) = 1 through a(11) = 9 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)
                                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)
                                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)
                                                  (5,3,1)  (5,4,1)    (9,2)
                                                           (7,2,1)    (5,4,2)
                                                           (4,3,2,1)  (6,3,2)
                                                                      (6,4,1)
                                                                      (7,3,1)
                                                                      (5,3,2,1)

The first condition alone gives A341450.
The non-strict version is A343346 (Heinz numbers: A343343).
The second condition alone gives A343377.
The strict complement is A343378.
The version for "and" instead of "or" is A343379.
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, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343337, A343338, A343341, A343342.

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

A339562 Squarefree numbers with no prime index dividing all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 51, 55, 69, 77, 85, 91, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 203, 205, 209, 215, 217, 219, 221, 231, 247, 249, 253, 255, 265, 285, 287, 291, 295, 299, 301, 309, 323, 327, 329, 335, 341, 345, 355, 357, 377, 381

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

First differs from A342193 in lacking 45.
Alternative name: 1 and squarefree numbers with smallest prime index not dividing 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 1 and Heinz numbers of strict integer partitions with smallest part not dividing all the others (counted by A341450). 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: {}         141: {2,15}     219: {2,21}
     15: {2,3}      143: {5,6}      221: {6,7}
     33: {2,5}      145: {3,10}     231: {2,4,5}
     35: {3,4}      155: {3,11}     247: {6,8}
     51: {2,7}      161: {4,9}      249: {2,23}
     55: {3,5}      165: {2,3,5}    253: {5,9}
     69: {2,9}      177: {2,17}     255: {2,3,7}
     77: {4,5}      187: {5,7}      265: {3,16}
     85: {3,7}      195: {2,3,6}    285: {2,3,8}
     91: {4,6}      201: {2,19}     287: {4,13}
     93: {2,11}     203: {4,10}     291: {2,25}
     95: {3,8}      205: {3,13}     295: {3,17}
    105: {2,3,4}    209: {5,8}      299: {6,9}
    119: {4,7}      215: {3,14}     301: {4,14}
    123: {2,13}     217: {4,11}     309: {2,27}

The squarefree complement is A339563.
These partitions are counted by A341450.
The not necessarily squarefree version is A342193.
A000005 counts divisors.
A000070 counts partitions with a selected part.
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).
A083710 counts partitions with a dividing part (strict: A097986).
Cf. A253249, A264401, A257993, A338470, A343337, A343338, A343339, A343379, A343380, A343382.

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

A130714 Number of partitions of n such that every part divides the largest part and such that the smallest part divides every part.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 27, 41, 42, 55, 64, 81, 83, 114, 116, 151, 168, 202, 210, 277, 289, 348, 382, 460, 478, 604, 623, 747, 812, 942, 1006, 1223, 1269, 1479, 1605, 1870, 1959, 2329, 2434, 2818, 3056, 3458, 3653, 4280, 4493, 5130, 5507, 6231, 6580

Offset: 1

Vladeta Jovovic, Jul 02 2007

First differs from A130689 at a(11) = 27, A130689(11) = 28.

Alternative name: Number of integer partitions of n with a part divisible by and a part dividing all the other parts. With this definition we have a(0) = 1. - Gus Wiseman, Apr 18 2021

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 though 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)

The second condition alone gives A083710.
The first condition alone gives A130689.
The opposite version is A343342.
The Heinz numbers of these partitions are the complement of A343343.
The half-opposite versions are A343344 and A343345.
The complement is counted by A343346.
The strict case is A343378.
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. A338470, A341450, A342193, A343337, A343338, A343341, A343379, A343382.

A130714 := proc(n) local gf,den,i,k,j ; gf := 0 ; for i from 0 to n do for j from 1 to n/(1+i) do den := 1 ; for k in numtheory[divisors](i) do den := den*(1-x^(j*k)) ; od ; gf := taylor(gf+x^(j+i*j)/den,x=0,n+1) ; od ; od: coeftayl(gf,x=0,n) ; end: seq(A130714(n),n=1..60) ; # R. J. Mathar, Oct 28 2007

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

G.f.: Sum_{i>=0} Sum_{j>0} x^(j+i*j)/Product_{k|i} (1-x^(j*k)).

More terms from R. J. Mathar, Oct 28 2007

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.

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}]

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.

cp's OEIS Frontend

A342193 Numbers with no prime index dividing all the other prime indices.

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

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A343382 Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.

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A339562 Squarefree numbers with no prime index dividing all the other prime indices.

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A130714 Number of partitions of n such that every part divides the largest part and such that the smallest part divides every part.

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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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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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