A018783 -id:A018783 - cp's OEIS frontend

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A304712 Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 25, 32, 43, 54, 70, 86, 105, 130, 162, 196, 240, 286, 339, 405, 485, 573, 674, 790, 922, 1072, 1252, 1456, 1685, 1939, 2226, 2557, 2923, 3349, 3822, 4347, 4931, 5593, 6335, 7170, 8092, 9105, 10233, 11495, 12903, 14458, 16169, 18063

Offset: 0

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1.

			The a(6) = 10 partitions whose parts are all equal or whose distinct parts are pairwise coprime are (6), (51), (411), (33), (321), (3111), (222), (2211), (21111), (111111).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709, A304711.

g:= proc(n, i, s) `if`(n=0, 1, `if`(i<1, 0,
      b(n, i, select(x-> x<=i, s))))
    end:
b:= proc(n, i, s) option remember; g(n, i-1, s)+(f->
     `if`(f intersect s={}, add(g(n-i*j, i-1, s union f)
        , j=1..n/i), 0))(numtheory[factorset](i))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2018

Table[Select[IntegerPartitions[n],Or[SameQ@@#,CoprimeQ@@Union[#]]&]//Length,{n,20}]
(* Second program: *)
g[n_, i_, s_] := If[n == 0, 1, If[i < 1, 0, b[n, i, Select[s, # <= i &]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + Function[f,
     If[f ~Intersection~ s == {}, Sum[g[n - i*j, i - 1, s ~Union~ f],
     {j, 1, n/i}], 0]][FactorInteger[i][[All, 1]]];
a[n_] := g[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A328170 Number of integer partitions of n whose parts minus 1 are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 8, 12, 18, 27, 38, 53, 74, 102, 137, 184, 241, 317, 413, 536, 687, 880, 1112, 1405, 1765, 2215, 2755, 3424, 4229, 5216, 6402, 7847, 9572, 11662, 14148, 17139, 20688, 24940, 29971, 35969, 43044, 51438, 61311, 72985, 86678, 102807, 121675

Offset: 0

Gus Wiseman, Oct 09 2019

nonn

A partition is relatively prime if the GCD of its parts is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(2) = 1 through a(9) = 18 partitions:
  (2)  (21)  (22)   (32)    (42)     (43)      (62)       (54)
             (211)  (221)   (222)    (52)      (332)      (63)
                    (2111)  (321)    (322)     (422)      (72)
                            (2211)   (421)     (431)      (432)
                            (21111)  (2221)    (521)      (522)
                                     (3211)    (2222)     (621)
                                     (22111)   (3221)     (3222)
                                     (211111)  (4211)     (3321)
                                               (22211)    (4221)
                                               (32111)    (4311)
                                               (221111)   (5211)
                                               (2111111)  (22221)
                                                          (32211)
                                                          (42111)
                                                          (222111)
                                                          (321111)
                                                          (2211111)
                                                          (21111111)

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

The Heinz numbers of these partitions are given by A328168.
Partitions whose parts are relatively prime are A000837.
Partitions whose parts plus 1 are relatively prime are A318980.
The GCD of the prime indices of n, all minus 1, is A328167(n).
Cf. A007359, A018783, A258409, A289509, A328163, A328164.

Table[Length[Select[IntegerPartitions[n],GCD@@(#-1)==1&]],{n,0,30}]

seq(n)=Vec(sum(d=1, n-1, moebius(d)*(-1/(1-x) + 1/prod(k=0, n\d, 1 - x*x^(k*d) + O(x*x^n)))), -(n+1)) \\ Andrew Howroyd, Oct 17 2019

G.f.: Sum_{d>=1} mu(d)*(-1/(1-x) + 1/(Prod_{k>=0} 1 - x^(k*d + 1))). - Andrew Howroyd, Oct 17 2019

A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(3) = 1 through a(11) = 13 partitions:
  (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
          (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
                        (5,2)             (6,3)      (3,3,2,2)  (7,4)
                        (3,2,2)           (7,2)                 (8,3)
                                          (3,3,3)               (9,2)
                                          (4,3,2)               (4,4,3)
                                          (5,2,2)               (5,4,2)
                                          (3,2,2,2)             (6,3,2)
                                                                (7,2,2)
                                                                (3,3,3,2)
                                                                (4,3,2,2)
                                                                (5,2,2,2)
                                                                (3,2,2,2,2)

This is the odd case of A018783, complement A000837.
The even version is A047967.
The complement is counted by A366850, ranks A366846.
A000041 counts integer partitions, strict A000009.
A000740 counts relatively prime compositions.
A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A289508 gives gcd of prime indices, positions of ones A289509.
Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

A366843 Number of integer partitions of n into odd, relatively prime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 21, 23, 32, 37, 42, 53, 62, 70, 88, 103, 116, 139, 164, 184, 220, 255, 283, 339, 390, 435, 511, 578, 653, 759, 863, 963, 1107, 1259, 1401, 1609, 1814, 2015, 2303, 2589, 2878, 3259, 3648, 4058, 4580, 5119, 5672, 6364

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (3111)    (511)      (71)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Allowing even parts gives A000837.
The strict case is A366844, with evens A078374.
The complement is counted by A366852, with evens A018783.
The pairwise coprime version is A366853, with evens A051424.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A113685, A116598, A289509, A289508, A302697, A337485, A366845, A366848, A366849.

Table[Length[Select[IntegerPartitions[n],#=={}||And@@OddQ/@#&&GCD@@#==1&]],{n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366843(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

A303139 Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 6, 13, 17, 33, 37, 68, 82, 125, 159, 237, 278, 409, 491, 674, 830, 1121, 1329, 1781, 2144, 2770, 3345, 4299, 5086, 6507, 7752, 9687, 11571, 14378, 16985, 21039, 24876, 30379, 35924, 43734, 51320, 62238, 73068, 87747, 103021, 123347, 143955

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(7) = 5 partitions are (421), (331), (322), (2221), (22111).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303140.

Table[Select[IntegerPartitions[n],!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,30}]

A303280 Number of strict integer partitions of n whose parts have a common divisor other than 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 10, 1, 11, 6, 12, 1, 19, 3, 18, 8, 23, 1, 36, 1, 32, 13, 38, 7, 57, 1, 54, 19, 68, 1, 95, 1, 90, 33, 104, 1, 148, 5, 149, 39, 166, 1, 230, 14, 226, 55, 256, 1, 360, 1, 340, 82, 390, 20, 527, 1, 513, 105, 609, 1

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

			The a(18) = 10 strict partitions are (18), (10,8), (12,6), (14,4), (15,3), (16,2), (8,6,4), (9,6,3), (10,6,2), (12,4,2).

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

Cf. A000009, A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303138.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> -add(mobius(d)*b(n/d), d=divisors(n) minus {1}):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 23 2018

Table[-Sum[MoebiusMu[d]*PartitionsQ[n/d],{d,Rest[Divisors[n]]}],{n,100}]

a(n) = -Sum_{d|n, d > 1} mu(d) * A000009(n/d).

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

Original entry on oeis.org

0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the column sums are not relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,1},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,2}}
                      {{1,2},{2,2}}
                      {{1,2},{3,3}}
                      {{1,2},{3,4}}
                      {{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
  {{1},{1}}  {{1},{1,1}}    {{1,1,2,2}}        {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1},{1,1,1}}      {{1,1},{1,1,1}}
                            {{1,1},{1,1}}      {{1},{1},{1,1,1}}
                            {{1},{1,2,2}}      {{1},{1,1},{1,1}}
                            {{1,1},{2,2}}      {{1},{1},{1},{1,1}}
                            {{1,2},{1,2}}      {{1},{1},{1},{1},{1}}
                            {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n));Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023

a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A328188 Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 12, 15, 15, 19, 23, 25, 30, 35, 39, 47, 52, 58, 65, 75, 86, 95, 109, 124, 144, 165, 181, 203, 221, 249, 285, 316, 352, 392, 438, 484, 538, 599, 666, 737, 813, 899, 992, 1102, 1215, 1335, 1472, 1621, 1776, 1946, 2137, 2336

Offset: 0

Gus Wiseman, Oct 13 2019

nonn

			The a(1) = 1 through a(15) = 15 partitions (A..F = 10..15):
  1  2  3   4   5   6    7   8    9    A     B     C    D     E     F
        21  31  32  51   43  53   54   73    65    75   76    95    87
                41  321  52  71   72   91    74    B1   85    B3    B4
                         61  431  81   532   83    543  94    D1    D2
                             521  432  541   92    651  A3    653   E1
                                  531  721   A1    732  B2    743   654
                                       4321  731   741  C1    752   753
                                             5321  831  652   761   852
                                                   921  751   851   951
                                                        832   941   A32
                                                        5431  A31   B31
                                                        7321  B21   6531
                                                              5432  7431
                                                              6521  7521
                                                              8321  54321

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The case of compositions is A167606.
The non-strict case is A328172.
The Heinz numbers of these partitions are given by A328335.
Partitions with no pairs of consecutive parts relatively prime are A328187.
Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328220.

b:= proc(n, i, s) option remember; `if`(i*(i+1)/2 igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
           numtheory[factorset](i)), 0)+b(n, i-1, s)))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 13 2019

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]>1]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[i(i + 1)/2 < n, 0, If[n == 0, 1, If[AllTrue[s,  GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1], FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A366844 Number of strict integer partitions of n into odd relatively prime parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 5, 4, 4, 5, 6, 7, 8, 8, 9, 11, 12, 12, 15, 16, 15, 19, 23, 23, 26, 28, 30, 34, 37, 38, 44, 48, 48, 56, 62, 63, 72, 77, 82, 92, 96, 102, 116, 124, 128, 142, 155, 162, 178, 191, 200, 222, 236, 246, 276, 291, 303, 334

Offset: 0

Gus Wiseman, Oct 29 2023

nonn

			The a(n) partitions for n = 1, 8, 14, 17, 16, 20, 21:
  (1)  (5,3)  (9,5)   (9,5,3)   (9,7)      (11,9)      (9,7,5)
       (7,1)  (11,3)  (9,7,1)   (11,5)     (13,7)      (11,7,3)
              (13,1)  (11,5,1)  (13,3)     (17,3)      (11,9,1)
                      (13,3,1)  (15,1)     (19,1)      (13,5,3)
                                (7,5,3,1)  (9,7,3,1)   (13,7,1)
                                           (11,5,3,1)  (15,5,1)
                                                       (17,3,1)

This is the relatively prime case of A000700.
The pairwise coprime version is the odd-part case of A007360.
Allowing even parts gives A078374.
The halved even version is A078374 aerated.
The non-strict version is A366843, with evens A000837.
The complement is counted by the strict case of A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A116598, A239261, A302697, A337485, A365067, A366845, A366848.

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#==1&]],{n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366844(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

More terms from Chai Wah Wu, Oct 30 2023

cp's OEIS Frontend

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

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A304712 Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.

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A328170 Number of integer partitions of n whose parts minus 1 are relatively prime.

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A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

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A366843 Number of integer partitions of n into odd, relatively prime parts.

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A303139 Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.

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A303280 Number of strict integer partitions of n whose parts have a common divisor other than 1.

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A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

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