A341450 -id:A341450 - 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

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).

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Gus Wiseman, Mar 23 2021

nonn

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

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (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)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

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

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
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.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

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}]

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

Original entry on oeis.org

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)]&]

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)]&]

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

cp's OEIS Frontend

A097986 Number of strict 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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A338470 Number of integer partitions of n with no part 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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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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A339563 Squarefree numbers > 1 whose smallest prime index divides 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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