A304714 -id:A304714 - cp's OEIS frontend

A305078 Heinz numbers of connected integer partitions.

2, 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

Offset: 1

Gus Wiseman, May 24 2018

nonn

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

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence lists all Heinz numbers of multisets S such that G(S) is a connected graph.

			The sequence of all connected multiset multisystems (see A302242, A112798) begins:
   2: {{}}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  59: {{7}}
  61: {{1,2,2}}
  63: {{1},{1},{1,1}}
  65: {{2},{1,2}}
  67: {{8}}
  71: {{1,1,3}}
  73: {{2,4}}
  79: {{1,5}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  97: {{3,3}}

Madeline Locus Dawsey, Tyler Russell and Dannie Urban, Polynomials Associated to Integer Partitions, arXiv:2108.00943 [math.NT], 2021.

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305079.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[300],Length[zsm[primeMS[#]]]==1&]

A304716 Number of integer partitions of n whose distinct parts are connected.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 3, 15, 4, 18, 12, 25, 11, 41, 17, 54, 36, 72, 44, 113, 69, 145, 113, 204, 153, 302, 220, 394, 343, 541, 475, 771, 662, 1023, 968, 1398, 1314, 1929, 1822, 2566, 2565, 3440, 3446, 4677, 4688, 6187, 6407, 8216, 8544, 10975, 11436

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(12) = 15 connected integer partitions and their corresponding connected multiset multisystems (see A112798, A302242) are the following.
                     (12): {{1,1,2}}
                    (6 6): {{1,2},{1,2}}
                    (8 4): {{1,1,1},{1,1}}
                    (9 3): {{2,2},{2}}
                   (10 2): {{1,3},{1}}
                  (4 4 4): {{1,1},{1,1},{1,1}}
                  (6 3 3): {{1,2},{2},{2}}
                  (6 4 2): {{1,2},{1,1},{1}}
                  (8 2 2): {{1,1,1},{1},{1}}
                (3 3 3 3): {{2},{2},{2},{2}}
                (4 4 2 2): {{1,1},{1,1},{1},{1}}
                (6 2 2 2): {{1,2},{1},{1},{1}}
              (4 2 2 2 2): {{1,1},{1},{1},{1},{1}}
            (2 2 2 2 2 2): {{1},{1},{1},{1},{1},{1}}
(1 1 1 1 1 1 1 1 1 1 1 1): {{},{},{},{},{},{},{},{},{},{},{},{}}

Cf. A000009, A003963, A048143, A054921, A218970, A285572, A286518, A302242, A304714, A305078, A305079, A322306, A322307.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[zsm[Union[#]]]===1&]],{n,30}]

For n > 1, a(n) = A218970(n) + 1. - Gus Wiseman, Dec 04 2018

Name changed to distinguish from A218970 by Gus Wiseman, Dec 04 2018

A305079 Number of connected components of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 5, 2, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 4, 1, 2, 1, 6, 1, 3, 1, 3, 2, 3, 1, 4, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 1, 3, 2, 2, 1

Offset: 1

Gus Wiseman, May 24 2018

nonn

First differs from |A305052(n)| at a(169) = 1, A305052(169) = 0.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. If S is the integer partition with Heinz number n, a(n) is the number of connected components of G(S).

			The a(315) = 2 connected components of {2,2,3,4} are {{3},{2,2,4}}.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305078, A305501.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[zsm[primeMS[n]]],{n,100}]

zero_first_elem_and_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2,#ys,if(ys[j]&&(1!=gcd(cs[i],ys[j])), listput(cs,ys[j]); ys[j] = 0)); i++); (ys); };
A007814(n) = valuation(n,2);
A000265(n) = (n/2^A007814(n));
A305079(n) = if(!(n%2),A007814(n)+A305079(A000265(n)), my(cs = apply(p -> primepi(p),factor(n)[,1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Nov 10 2018

For all n, k > 0, we have a(2^n * k) = n + a(k).
For all x, y > 0, we have a(x * y) <= a(x) + a(y).
For x, y > 0 strongly coprime, we have a(x * y) = a(x) + a(y). Strongly coprime means every prime index of x is coprime to every prime index of y, where a prime index of n is a number m such that prime(m) divides n.
a(n) = A305501(A064989(n)) + A007814(n). - Antti Karttunen, Nov 10 2018

Terms and Mathematica program corrected by Gus Wiseman, Nov 10 2018

A007360 Number of partitions of n into distinct and pairwise relatively prime parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 9, 10, 11, 10, 13, 17, 19, 21, 22, 21, 24, 32, 37, 37, 38, 40, 45, 55, 65, 69, 66, 64, 75, 86, 100, 113, 107, 106, 122, 145, 165, 174, 167, 162, 179, 222, 253, 255, 255, 255, 273, 328, 373, 376, 369, 377, 406, 476, 553, 569, 537, 529

Offset: 1

N. J. A. Sloane and Mira Bernstein, following a suggestion from Marc LeBrun

			From _Gus Wiseman_, Sep 23 2019: (Start)
The a(1) = 1 through a(10) = 6 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)   (8)    (9)    (A)
            (21)  (31)  (32)  (51)   (43)  (53)   (54)   (73)
                        (41)  (321)  (52)  (71)   (72)   (91)
                                     (61)  (431)  (81)   (532)
                                           (521)  (531)  (541)
                                                         (721)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..750 (terms 1..350 from Alois P. Heinz)
M. LeBrun & D. Hoey, Emails

Number of partitions of n into relatively prime parts = A000837.
The non-strict case is A051424.
Strict relatively prime partitions are A078374.
Cf. A007359, A038348, A084422, A186974, A187106, A303140, A302569, A303362, A304714, A320426, A320436.

$RecursionLimit = 1000; b[n_, i_, s_] := b[n, i, s] = Module[{f}, If[n == 0 || i == 1, 1, If[i<2, 0, f = FactorInteger[i][[All, 1]]; b[n, i-1, Select[s, #Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]==1||UnsameQ@@#&&CoprimeQ@@Union[#]&]],{n,0,30}] (* Gus Wiseman, Sep 23 2019 *)

a(n) = A051424(n)-A051424(n-2). - Vladeta Jovovic, Dec 11 2004

More precise definition from Vladeta Jovovic, Dec 11 2004

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005

A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

Original entry on oeis.org

0, -1, -1, -2, -1, -2, -1, -3, -1, -2, -1, -3, -1, -2, -2, -4, -1, -2, -1, -3, -1, -2, -1, -4, -1, -2, -1, -3, -1, -3, -1, -5, -2, -2, -2, -3, -1, -2, -1, -4, -1, -2, -1, -3, -2, -2, -1, -5, -1, -2, -2, -3, -1, -2, -2, -4, -1, -2, -1, -4, -1, -2, -1, -6, -1, -3

Offset: 1

Gus Wiseman, May 24 2018

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.
First nonnegative entry after a(1) = 0 is a(169) = 0.

			The 1105th multiset multisystem is {{2},{1,2},{4}} with clutter density -2, so a(1105) = -2.
The 5429th multiset multisystem is {{1,2,2},{1,1,1,2}} with clutter density 0, so a(5429) = 0.
The 11837th multiset multisystem is {{1,1},{1,1,1},{1,1,1,2}} with clutter density -1, so a(11837) = -1.
The 42601th multiset multisystem is {{1,2},{1,3},{1,2,3}} with clutter density 1, so a(42601) = 1.

Cf. A001221, A030019, A048143, A056239, A112798, A285572, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A304911, A305001.

zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Array[zens,100]

A322338 Edge-connectivity of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

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

The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			2093 is the Heinz number of (9,6,4), corresponding to the multiset partition {{1,1},{1,2},{2,2}}, which can be made disconnected by removing only the part {1,2}, so a(2093) = 1.

Wikipedia, k-edge-connected graph

Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A302242, A304714, A304716, A305078, A305079, A322335, A322336, A322337.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[PrimeOmega[n]-Max@@PrimeOmega/@Select[Divisors[n],Length[csm[primeMS/@primeMS[#]]]!=1&],{n,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

A328514 MM-numbers of connected sets of sets.

Original entry on oeis.org

1, 2, 3, 5, 11, 13, 17, 29, 31, 39, 41, 43, 47, 59, 65, 67, 73, 79, 83, 87, 101, 109, 113, 127, 129, 137, 139, 149, 157, 163, 167, 179, 181, 191, 195, 199, 211, 233, 235, 237, 241, 257, 269, 271, 277, 283, 293, 303, 313, 317, 319, 331, 339, 347, 349, 353, 365

Offset: 1

Gus Wiseman, Oct 20 2019

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence all connected set of sets together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  29: {{1,3}}
  31: {{5}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  59: {{7}}
  65: {{2},{1,2}}
  67: {{8}}
  73: {{2,4}}
  79: {{1,5}}
  83: {{9}}
  87: {{1},{1,3}}

The not-necessarily-connected case is A302494.
BII-numbers of connected set-systems are A326749.
MM-numbers of connected sets of multisets are A328513.
Cf. A005117, A007947, A056239, A112798, A286518, A302242, A302569, A304714, A305078, A305079, A327398.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Select[Range[1000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]

Intersection of A302494 and A305078.

A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 9, 5, 9, 8, 13, 10, 15, 9, 15, 13, 18, 14, 22, 21, 26, 19, 29, 24, 36, 31, 40, 35, 45, 38, 54, 55, 59, 55, 70, 69, 84, 74, 89, 86, 107, 103, 119, 115, 143, 143, 159

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.
         (34): {{1,7}}
       (30 4): {{1,2,3},{1,1}}
       (28 6): {{1,1,4},{1,2}}
       (26 8): {{1,6},{1,1,1}}
      (24 10): {{1,1,1,2},{1,3}}
      (22 12): {{1,5},{1,1,2}}
      (20 14): {{1,1,3},{1,4}}
     (20 8 6): {{1,1,3},{1,1,1},{1,2}}
      (18 16): {{1,2,2},{1,1,1,1}}
    (15 10 9): {{2,3},{1,3},{2,2}}
   (15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}
    (14 12 8): {{1,4},{1,1,2},{1,1,1}}
  (14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}

Cf. A000009, A003963, A006126, A048143, A051424, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A293993, A302242, A303362, A304714, A304716.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,30}]

A322335 Number of 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 1, 37, 2, 45, 18, 58, 8, 95, 19, 109, 45, 150, 38, 232, 59, 268, 129, 357, 155, 523, 203, 633, 359, 852, 431, 1185, 609, 1464, 969

Offset: 1

Gus Wiseman, Dec 04 2018

First differs from A108572 at a(17) = 1, A108572(17) = 0.

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part. For example (6,6,3,2) is 2-edge-connected but (6,3,2) is not.

			The a(14) = 15 2-edge-connected integer partitions of 14:
  (7,7)   (6,4,4)   (4,4,4,2)  (4,4,2,2,2)  (4,2,2,2,2,2)  (2,2,2,2,2,2,2)
  (8,6)   (6,6,2)   (6,4,2,2)  (6,2,2,2,2)
  (10,4)  (8,4,2)   (8,2,2,2)
  (12,2)  (10,2,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A322336, A322337, A322338.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],twoedQ[primeMS/@#]&]],{n,30}]

a(42)-a(45) from Jinyuan Wang, Jun 20 2020

cp's OEIS Frontend

A305078 Heinz numbers of connected integer partitions.

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A304716 Number of integer partitions of n whose distinct parts are connected.

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A305079 Number of connected components of the integer partition with Heinz number n.

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A007360 Number of partitions of n into distinct and pairwise relatively prime parts.

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A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

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A322338 Edge-connectivity of the integer partition with Heinz number n.

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

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A328514 MM-numbers of connected sets of sets.

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