A343344 -id:A343344 - cp's OEIS frontend

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

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

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)]&]

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

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}]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

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

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

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

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

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

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

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

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}]

A343339 Numbers with no prime index dividing all the other prime indices, but with a prime index divisible by all the other prime indices.

Original entry on oeis.org

195, 555, 585, 915, 957, 975, 1295, 1335, 1665, 1695, 1755, 2193, 2265, 2343, 2535, 2585, 2715, 2745, 2775, 2871, 2925, 3115, 3345, 3367, 3729, 3765, 3885, 4005, 4209, 4215, 4575, 4755, 4875, 4995, 5085, 5265, 5285, 5385, 5457, 5467, 5709, 5955, 6205, 6215

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Numbers > 1 whose smallest prime index does not divide all the other prime indices, but whose greatest prime index is 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 with greatest part divisible by all the others, but smallest part not dividing all the others (counted by A343344). 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:
     195: {2,3,6}        2585: {3,5,15}       4575: {2,3,3,18}
     555: {2,3,12}       2715: {2,3,42}       4755: {2,3,66}
     585: {2,2,3,6}      2745: {2,2,3,18}     4875: {2,3,3,3,6}
     915: {2,3,18}       2775: {2,3,3,12}     4995: {2,2,2,3,12}
     957: {2,5,10}       2871: {2,2,5,10}     5085: {2,2,3,30}
     975: {2,3,3,6}      2925: {2,2,3,3,6}    5265: {2,2,2,2,3,6}
    1295: {3,4,12}       3115: {3,4,24}       5285: {3,4,36}
    1335: {2,3,24}       3345: {2,3,48}       5385: {2,3,72}
    1665: {2,2,3,12}     3367: {4,6,12}       5457: {2,7,28}
    1695: {2,3,30}       3729: {2,5,30}       5467: {4,5,20}
    1755: {2,2,2,3,6}    3765: {2,3,54}       5709: {2,5,40}
    2193: {2,7,14}       3885: {2,3,4,12}     5955: {2,3,78}
    2265: {2,3,36}       4005: {2,2,3,24}     6205: {3,7,21}
    2343: {2,5,20}       4209: {2,9,18}       6215: {3,5,30}
    2535: {2,3,6,6}      4215: {2,3,60}       6475: {3,3,4,12}

The first condition alone gives A342193.
The second condition alone gives the complement of A343337.
The partitions with these Heinz numbers are counted by A343344.
A000005 counts divisors.
A000070 counts partitions with a selected part.
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. A130689, A130714, A257993, A338470, A339562, A341450, A343338, A343380.

Select[Range[2,1000],With[{p=PrimePi/@First/@FactorInteger[#]},And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]

Complement of A343337 in A342193.

A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 5, 2, 5, 5, 4, 5, 7, 3, 5, 6, 5, 5, 9, 4, 7, 6, 6, 9, 11, 6, 9, 10, 9, 10, 12, 6, 11, 12, 10, 12, 16, 9, 15, 16, 12, 14, 18, 14, 16, 18, 14, 15, 22, 11, 16, 20, 13, 21, 23, 15, 21, 24, 21, 21, 31, 14, 24

Offset: 0

Gus Wiseman, Apr 19 2021

nonn

Alternative name: Number of strict integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

			The a(n) partitions for n = 14, 12, 18, 24, 30, 39, 36:
  (14)     (12)    (18)      (24)        (30)        (39)          (36)
  (12,2)   (8,4)   (12,6)    (16,8)      (24,6)      (36,3)        (27,9)
  (8,4,2)  (9,3)   (15,3)    (18,6)      (25,5)      (26,13)       (30,6)
           (10,2)  (16,2)    (20,4)      (27,3)      (27,9,3)      (32,4)
                   (12,4,2)  (21,3)      (28,2)      (28,7,4)      (33,3)
                             (22,2)      (20,10)     (30,6,3)      (34,2)
                             (12,6,4,2)  (18,9,3)    (24,12,3)     (24,12)
                                         (24,4,2)    (24,8,4,3)    (24,8,4)
                                         (16,8,4,2)  (20,10,5,4)   (18,9,6,3)
                                                     (24,6,4,3,2)  (24,6,4,2)
                                                                   (20,10,4,2)

The dual version is A098965 (non-strict: A083711).
The non-strict version is A339619 (Heinz numbers: complement of A343337).
The version with 1's allowed is A343347 (non-strict: A130689).
The case without a part dividing all the other parts is A343380.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A083710, A097986, A098743, A200745, A264401, A339563, A341450, A343337, A343341, A343344, A343378.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]

cp's OEIS Frontend

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

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A342193 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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A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

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

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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.

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

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A343339 Numbers with no prime index dividing all the other prime indices, but with a prime index divisible by all the other prime indices.

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