A018783 -id:A018783 - cp's OEIS frontend

A318978 Heinz numbers of integer partitions with a common divisor > 1.

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

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

Is this the same as A305078 without the leading 2? - R. J. Mathar, Sep 08 2018

			The sequence of all integer partitions with a common divisor begins: (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9), (3,3), (2,2,2), (10), (11), (12), (6,2), (13), (14), (15), (4,4), (16), (8,2), (17), (18), (4,2,2), (6,3), (19), (20), (21), (22), (2,2,2,2), (23), (10,2), (24), (6,4), (25).

Cf. A000837, A018783, A056239, A289508, A289509, A296150, A298748.

Select[Range[100],GCD@@PrimePi/@If[#==1,{},FactorInteger[#]][[All,1]]>1&]

A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 24, 28, 36, 43, 51, 62, 74, 87, 104, 122, 143, 169, 195, 227, 260, 302, 346, 397, 455, 521, 599, 686, 780, 889, 1001, 1138, 1286, 1454, 1638, 1846, 2076, 2330, 2614, 2929, 3280, 3666, 4093, 4565, 5085, 5667, 6300, 7002

Offset: 0

Gus Wiseman, Oct 12 2019

nonn

Except for any number of 1's, these partitions must be strict. The fully strict case is A328188.

Partitions with no consecutive pair of parts relatively prime are A328187, with strict case A328220.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

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

The case of compositions is A167606.
The strict case is A328188.
The Heinz numbers of these partitions are given by A328335.
Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328188 A328220.

b:= proc(n, i, s) option remember; `if`(n=0 or i=1, 1,
      `if`(andmap(j-> 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],!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]>1]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0 || i == 1, 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 *)

A304709 Number of integer partitions of n whose distinct parts are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 13, 16, 23, 29, 42, 49, 69, 83, 102, 126, 161, 191, 239, 281, 336, 402, 484, 566, 672, 787, 919, 1067, 1251, 1449, 1684, 1934, 2223, 2554, 2920, 3341, 3821, 4344, 4928, 5586, 6334, 7163, 8091, 9100, 10228, 11492, 12902, 14449, 16167, 18058

Offset: 1

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

			The a(6) = 7 integer partitions of 6 whose distinct parts are pairwise coprime are (51), (411), (321), (3111), (2211), (21111), (111111).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000005, A007359, A007360, A018783, A051424, A078374, A101268, A289508, A289509, A302569, A302696, A302698, A302796, A302797, A304711, A304712.

Table[Select[IntegerPartitions[n],CoprimeQ@@Union[#]&]//Length,{n,20}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sum(i=1, n-1, if(gcd(i,n)>1, 2^(n-i)))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, 0) + 1 - numdiv(n));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

a(n) = A304712(n) + 1 - A000005(n). - Andrew Howroyd, Nov 02 2019

A328187 Number of integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 3, 38, 4, 46, 19, 58, 9, 94, 18, 106, 41, 144, 28, 221, 37, 246, 92, 318, 87, 465, 95, 530, 198, 693, 169, 963, 220, 1108, 424, 1383, 381, 1899, 492, 2216, 815, 2732, 799, 3644, 1041, 4231, 1585, 5194, 1608

Offset: 0

Gus Wiseman, Oct 12 2019

nonn

			The a(1) = 1 through a(15) = 9 partitions (A..F = 10..15):
  1  2  3  4   5  6    7  8     9    A      B  C       D  E        F
           22     33      44    63   55        66         77       96
                  42      62    333  64        84         86       A5
                  222     422        82        93         A4       C3
                          2222       442       A2         C2       555
                                     622       444        644      663
                                     4222      633        662      933
                                     22222     642        842      6333
                                               822        A22      33333
                                               3333       4442
                                               4422       6422
                                               6222       8222
                                               42222      44222
                                               222222     62222
                                                          422222
                                                          2222222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

The Heinz numbers of these partitions are given by A328336.
The case of compositions is A178470.
The strict case is A328220.
Partitions with all pairs of consecutive parts relatively prime are A328172.
Cf. A000837, A018783, A328028, A328170, A328171, A328188.

Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{n,0,30}]

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 9, 5, 10, 0, 16, 2, 14, 7, 17, 0, 27, 1, 21, 11, 24, 6, 36, 1, 30, 15, 37, 2, 51, 1, 41, 25, 44, 2, 64, 5, 58, 25, 57, 2, 81, 13, 69, 31, 70, 3, 108, 5, 80, 43, 85, 17, 123, 5, 97, 46, 120, 6, 144, 6

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A082024 at a(31) = 1, A082024(31) = 0.

The first relatively prime triple is (15,10,6), counted under a(31).

			The a(6) = 1 through a(16) = 5 partitions are (empty columns indicated by dots, A..G = 10..16):
  222  .  422  333  442  .  444  .  644  555  664  .  666  .  866
                    622     633     662  663  844     864     884
                            642     842  933  862     882     A55
                            822     A22       A42     963     A64
                                              C22     A44     A82
                                                      A62     C44
                                                      C33     C62
                                                      C42     E42
                                                      E22     G22

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014612 intersected with A337694 ranks these partitions.
A200976 and A328673 count these partitions of any length.
A284825 is the case that is also relatively prime.
A307719 is the pairwise coprime instead of non-coprime version.
A335402 gives the positions of zeros.
A337604 is the ordered version.
A337605 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000212, A000217, A001840, A018783, A082024, A211540, A220377, A337461.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]

A218970 Number of connected cyclic conjugacy classes of subgroups of the symmetric group.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 3, 8, 2, 14, 3, 17, 11, 24, 10, 40, 16, 53, 35, 71, 43, 112, 68, 144, 112, 203, 152, 301, 219, 393, 342, 540, 474, 770, 661, 1022, 967, 1397, 1313, 1928, 1821, 2565, 2564, 3439, 3445, 4676, 4687, 6186, 6406, 8215, 8543, 10974, 11435

Offset: 0

Liam Naughton, Nov 26 2012

nonn

a(n) is also the number of connected partitions of n in the following sense. Given a partition of n, the vertices are the parts of the partition and two vertices are connected if and only if their gcd is greater than 1. We call a partition connected if the graph is connected.

			From _Gus Wiseman_, Dec 03 2018: (Start)
The a(12) = 14 connected integer partitions of 12:
  (12)  (6,6)   (4,4,4)  (3,3,3,3)  (4,2,2,2,2)  (2,2,2,2,2,2)
        (8,4)   (6,3,3)  (4,4,2,2)
        (9,3)   (6,4,2)  (6,2,2,2)
        (10,2)  (8,2,2)
(End)

Liam Naughton and Goetz Pfeiffer, Integer sequences realized by the subgroup pattern of the symmetric group, arXiv:1211.1911 [math.GR], 2012-2013.
Liam Naughton, CountingSubgroups.g
Liam Naughton and Goetz Pfeiffer, Tomlib, The GAP table of marks library

Cf. A018783, A200976, A286518, A286520, A290103, A304714, A304716, A305078, A305079, A322306, A322307.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]==1&]],{n,10}]

For n > 1, a(n) = A304716(n) - 1. - Gus Wiseman, Dec 03 2018

More terms from Gus Wiseman, Dec 03 2018

A303140 Number of strict 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, 0, 0, 1, 0, 2, 2, 4, 2, 8, 7, 14, 14, 21, 18, 33, 32, 50, 54, 72, 67, 103, 110, 145, 155, 201, 196, 271, 293, 372, 400, 493, 512, 647, 704, 858, 924, 1115, 1167, 1436, 1560, 1854, 2022, 2368, 2510, 3005, 3255, 3804, 4144, 4792, 5116, 5989, 6514, 7486

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(14) = 7 partitions are (932), (8321), (7421), (653), (6521), (6431), (5432).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303139.

Table[Select[IntegerPartitions[n],UnsameQ@@#&&!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,20}]

A319149 Number of superperiodic integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 5, 1, 7, 1, 7, 3, 3, 1, 13, 2, 3, 4, 9, 1, 13, 1, 11, 3, 3, 3, 23, 1, 3, 3, 20, 1, 17, 1, 16, 9, 3, 1, 38, 2, 9, 3, 23, 1, 25, 3, 36, 3, 3, 1, 71, 1, 3, 11, 49, 3, 31, 1, 52, 3, 19

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.

			The a(24) = 11 superperiodic partitions:
  (24)
  (12,12)
  (8,8,8)
  (9,9,3,3)
  (8,8,4,4)
  (6,6,6,6)
  (10,10,2,2)
  (6,6,6,2,2,2)
  (6,6,4,4,2,2)
  (4,4,4,4,4,4)
  (4,4,4,4,2,2,2,2)
  (3,3,3,3,3,3,3,3)
  (2,2,2,2,2,2,2,2,2,2,2,2)

Cf. A000041, A000837, A018783, A024994, A047966, A098859, A100953, A181819, A182857, A304660, A305563, A317245, A317256, A319151, A319152, A319157.

wotperQ[m_]:=Or[m=={1},And[GCD@@m>1,wotperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],wotperQ]],{n,30}]

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

A partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are distinct, so y is counted under A239455(9).
A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).
The Heinz numbers of these partitions are given by A351294 \ A130091.
Is a(17) = 0 the last zero of the sequence?

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Wilf partitions are counted by A098859, ranked by A130091.
Look-and-Say partitions are counted by A239455, ranked by A351294.
Non-Wilf partitions are counted by A336866, ranked by A130092.
Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
A000569 = number of graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A225485/A325280 = frequency depth, ranked by A182850/A323014.
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths
A351013 = compositions with all distinct runs.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

More terms from Jinyuan Wang, Feb 14 2025

A328220 Number of strict integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 3, 10, 3, 11, 7, 12, 3, 19, 5, 18, 12, 23, 9, 36, 11, 33, 21, 40, 20, 58, 19, 58, 35, 70, 31, 98, 36, 101, 65, 112, 56, 155, 64, 164, 97, 188, 88, 250, 112, 256, 157, 293, 145, 392, 163, 399, 241, 461, 242

Offset: 0

Gus Wiseman, Oct 14 2019

nonn

			The a(2) = 1 through a(20) = 11 partitions (A..K = 10..20):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F   G    H    I    J    K
              42     62  63  64     84      86   96  A6   863  A8   964  C8
                             82     93      A4   A5  C4   962  C6   A63  E6
                                    A2      C2   C3  E2        E4        F5
                                    642     842      862       F3        G4
                                                     A42       G2        I2
                                                               864       A64
                                                               963       A82
                                                               A62       C62
                                                               C42       E42
                                                                         8642

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

The non-strict case is A328187.
Partitions with all consecutive parts relatively prime are A328172, with strict case A328188.
Strict partitions with relatively prime parts are A078374.
Partitions with no consecutive divisibilities are A328171.
Cf. A000837, A018783, A070211, A289509, A318980, A325545, A328170, A328221, A328335, A328336.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{n,0,30}]

cp's OEIS Frontend

A318978 Heinz numbers of integer partitions with a common divisor > 1.

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A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime.

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A304709 Number of integer partitions of n whose distinct parts are pairwise coprime.

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A328187 Number of integer partitions of n with no pair of consecutive parts relatively prime.

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A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

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A218970 Number of connected cyclic conjugacy classes of subgroups of the symmetric group.

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A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

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A319149 Number of superperiodic integer partitions of n.

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