A306017 -id:A306017 - cp's OEIS frontend

A320325 Numbers whose product of prime indices is a perfect power.

7, 9, 14, 18, 19, 21, 23, 25, 27, 28, 36, 38, 42, 46, 49, 50, 53, 54, 56, 57, 63, 72, 76, 81, 84, 92, 97, 98, 100, 103, 106, 108, 112, 114, 115, 121, 125, 126, 131, 133, 144, 147, 151, 152, 159, 161, 162, 168, 169, 171, 175, 183, 184, 185, 189, 194, 195, 196

Offset: 1

Gus Wiseman, Oct 10 2018

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 terms together with their corresponding multiset multisystems (A302242):
   7: {{1,1}}
   9: {{1},{1}}
  14: {{},{1,1}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}
  42: {{},{1},{1,1}}
  46: {{},{2,2}}
  49: {{1,1},{1,1}}
  50: {{},{2},{2}}
  53: {{1,1,1,1}}
  54: {{},{1},{1},{1}}
  56: {{},{},{},{1,1}}
  57: {{1},{1,1,1}}
  63: {{1},{1},{1,1}}
  72: {{},{},{},{1},{1}}
  76: {{},{},{1,1,1}}
  81: {{1},{1},{1},{1}}

Cf. A000720, A001222, A001597, A003963, A056239, A064573, A112798, A302242, A305551, A306017, A319056, A319066, A320322, A320323, A320324.

Select[Range[100],GCD@@FactorInteger[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]][[All,2]]>1&]

A000665 Number of 3-uniform hypergraphs on n unlabeled nodes, or equivalently number of relations with 3 arguments on n nodes.

Original entry on oeis.org

1, 1, 1, 2, 5, 34, 2136, 7013320, 1788782616656, 53304527811667897248, 366299663432194332594005123072, 1171638318502989084030402509596875836036608, 3517726593606526072882013063011594224625680712384971214848

Offset: 0

N. J. A. Sloane

The Qian reference has one incorrect term. The formula given in corollary 2.6 also contains a minor error. The second summation needs to be over p_i*p_j*p_h/lcm(p_i, p_j, p_h) rather than gcd(p_i, p_j, p_h)^2. - Andrew Howroyd, Dec 11 2018

			From _Gus Wiseman_, Dec 13 2018: (Start)
Non-isomorphic representatives of the a(5) = 34 hypergraphs:
  {}
  {{123}}
  {{125}{345}}
  {{134}{234}}
  {{123}{245}{345}}
  {{124}{134}{234}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{123}{124}{134}{234}}
  {{123}{145}{245}{345}}
  {{124}{135}{245}{345}}
  {{125}{135}{245}{345}}
  {{134}{235}{245}{345}}
  {{145}{235}{245}{345}}
  {{123}{124}{135}{245}{345}}
  {{123}{145}{235}{245}{345}}
  {{124}{134}{235}{245}{345}}
  {{134}{145}{235}{245}{345}}
  {{135}{145}{235}{245}{345}}
  {{145}{234}{235}{245}{345}}
  {{123}{124}{134}{235}{245}{345}}
  {{123}{134}{145}{235}{245}{345}}
  {{123}{145}{234}{235}{245}{345}}
  {{124}{135}{145}{235}{245}{345}}
  {{125}{135}{145}{235}{245}{345}}
  {{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{235}{245}{345}}
  {{124}{135}{145}{234}{235}{245}{345}}
  {{125}{135}{145}{234}{235}{245}{345}}
  {{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{234}{235}{245}{345}}
  {{125}{134}{135}{145}{234}{235}{245}{345}}
  {{124}{125}{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}
(End)

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..28 (first 26 terms from Andrew Howroyd)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Victor Falgas-Ravry and Emil R. Vaughan, On applications of Razborov's flag algebra calculus to extremal 3-graph theory, arXiv preprint arXiv:1110.1623 [math.CO], 2011.
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
E. M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390.
Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989.

Row sums of A092337. Spanning 3-uniform hypergraphs are counted by A322451.
Cf. A000088, A006129, A038041, A299471, A301922, A302374, A302394, A306017, A306021, A320395.
Column k=3 of A309858.

(* about 85 seconds on a laptop computer *)
Needs["Combinatorica`"];Table[A = Subsets[Range[n],{3}];CycleIndex[Replace[Map[Sort,System`PermutationReplace[A, SymmetricGroup[n]], {2}],Table[A[[i]] -> i, {i, 1, Length[A]}], 2], s] /. Table[s[i] -> 2, {i, 1, Binomial[n, 3]}], {n, 1, 8}] (* Geoffrey Critzer, Oct 28 2015 *)
Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Subsets[Range[n],{3}]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length],{prm,Permutations[Range[n]]}]/n!,{n,8}] (* Gus Wiseman, Dec 13 2018 *)
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[p_] := Sum[Ceiling[(p[[i]] - 1)*((p[[i]] - 2)/6)], {i, 1, Length[p]}] + Sum[Sum[c = p[[i]]; d = p[[j]]; GCD[c, d]*(c + d - 2 + Mod[(c - d)/GCD[c, d], 2])/2 + Sum[c*d*p[[k]]/LCM[c, d, p[[k]]], {k, 1, j - 1}], {j, 1, i - 1}], {i, 2, Length[p]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
a /@ Range[0, 12] (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(p)={sum(i=1, #p, ceil((p[i]-1)*(p[i]-2)/6)) + sum(i=2, #p, sum(j=1, i-1, my(c=p[i], d=p[j]); gcd(c,d)*(c + d - 2 + (c-d)/gcd(c,d)%2)/2 + sum(k=1, j-1, c*d*p[k]/lcm(lcm(c,d), p[k]))))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Dec 11 2018

Corrected and extended by Vladeta Jovovic

a(0)=1 prepended and a(12) from Andrew Howroyd, Dec 11 2018

A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 8, 30, 32, 342, 128, 3754, 11360, 56138, 2048, 3834670, 8192, 27528494, 577439424, 2681075210, 131072, 238060300946, 524288, 11045144602614, 115488471132032, 49840258213638, 8388608, 152185891301461434, 140102945910265344, 124260001149229146, 85092642310351607968

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

			The a(3) = 8 multiset partitions:
  {{1,1,1}}
  {{1,1,2}}
  {{1,2,2}}
  {{1,2,3}}
  {{1},{1},{1}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A255906, A298422, A306017, A306019, A306020, A306021, A320324, A322794, A326517, A326518, A326519, A326520, A326521, A331315.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Length/@#&]],{n,8}]

\\ here U(n,m) gives number for m blocks of size n.
U(n,m)={sum(k=1, n*m, binomial(binomial(k+n-1, n)+m-1, m)*sum(r=k, n*m, binomial(r, k)*(-1)^(r-k)) )}
a(n)={sumdiv(n, d, U(d, n/d))} \\ Andrew Howroyd, Sep 15 2018

a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} A331315(n/d, d). - Andrew Howroyd, Jan 15 2020

Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018

A319190 Number of regular hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 3, 19, 879, 5280907, 1069418570520767

Offset: 0

Gus Wiseman, Dec 17 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

			The a(3) = 19 regular hypergraphs:
                 {{1,2,3}}
                {{1},{2,3}}
                {{2},{1,3}}
                {{3},{1,2}}
               {{1},{2},{3}}
            {{1},{2,3},{1,2,3}}
            {{2},{1,3},{1,2,3}}
            {{3},{1,2},{1,2,3}}
            {{1,2},{1,3},{2,3}}
           {{1},{2},{3},{1,2,3}}
           {{1},{2},{1,3},{2,3}}
           {{1},{3},{1,2},{2,3}}
           {{2},{3},{1,2},{1,3}}
        {{1,2},{1,3},{2,3},{1,2,3}}
       {{1},{2},{1,3},{2,3},{1,2,3}}
       {{1},{3},{1,2},{2,3},{1,2,3}}
       {{2},{3},{1,2},{1,3},{1,2,3}}
      {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Column sums of A188445.

Cf. A002829, A005176, A049311, A058891, A110100, A110101, A116539, A283877, A295193, A306017, A319189.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{1,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,2^n}],{n,5}]

a(6) from Andrew Howroyd, Mar 12 2020

A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 26 2018

nonn

Also the number of uniform multiset partitions of the multiset of prime indices of n, where a multiset partition is uniform if all parts have the same size.

			The a(1260) = 13 factorizations:
  (1260)  (18*70)   (4*9*35)   (2*2*3*3*5*7)
          (20*63)   (6*6*35)
          (28*45)   (4*15*21)
          (30*42)   (6*10*21)
          (12*105)  (6*14*15)
                    (9*10*14)
The a(1260) = 13 multiset partitions:
  {{1},{1},{2},{2},{3},{4}}
     {{1,1},{2,2},{3,4}}
     {{1,1},{2,3},{2,4}}
     {{1,2},{1,2},{3,4}}
     {{1,2},{1,3},{2,4}}
     {{1,2},{1,4},{2,3}}
     {{2,2},{1,3},{1,4}}
      {{1,1,2},{2,3,4}}
      {{1,2,2},{1,3,4}}
      {{1,1,3},{2,2,4}}
      {{1,1,4},{2,2,3}}
      {{1,2,3},{1,2,4}}
       {{1,1,2,2,3,4}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001055, A001222, A038041, A112798, A306017, A306021, A319169, A320324, A317583, A321455, A321469, A326514, A326515, A326516.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@PrimeOmega/@#&]],{n,100}]

A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 41, 11, 1, 1, 768, 958, 26, 1, 1, 27449, 1042642, 32596, 57, 1, 1, 1887284, 34352419335, 34359509614, 2096731, 120, 1, 1, 252522481, 72057319189324805, 1180591620442534312297, 72057594021152435, 268434467, 247, 1, 1, 66376424160

Offset: 1

Gus Wiseman, Jun 18 2018

			Triangle begins:
  1;
  1,     1;
  1,     4,       1;
  1,    41,      11,     1;
  1,   768,     958,    26,  1;
  1, 27449, 1042642, 32596, 57, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..91 (rows 1..13)
Wikipedia, Hypergraph

Columns 1..4 are A000012, A006129, A302374, A302396.
Row sums are A306021.
The unlabeled version is A301922.
The connected version is A299354.
Cf. A000005, A001315, A006126, A038041, A298422, A298426, A306017, A306018, A306019, A306020.

Table[Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,10},{k,n}]

T(n, k) = sum(d = 0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)) \\ Andrew Howroyd, Jan 16 2024

T(n, k) = Sum_{d = 0..n} (-1)^(n-d)*binomial(n,d)*2^binomial(d,k).

A326512 Number of set partitions of {1..n} where every block has the same average.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 18, 16, 75, 64, 405, 302, 2581, 1693, 19872, 11295, 175807, 87524, 1851135, 787515, 21909766, 8185713, 298698113, 96514608, 4538610230, 1285072142

Offset: 0

Gus Wiseman, Jul 11 2019

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024

			The a(1) = 1 through a(7) = 18 set partitions:
  {1}  {12}  {123}    {1234}    {12345}      {123456}      {1234567}
             {13}{2}  {14}{23}  {1245}{3}    {1256}{34}    {123567}{4}
                                {135}{24}    {1346}{25}    {12467}{35}
                                {15}{234}    {16}{2345}    {1267}{345}
                                {15}{24}{3}  {16}{25}{34}  {13457}{26}
                                                           {1357}{246}
                                                           {1456}{237}
                                                           {147}{2356}
                                                           {156}{2347}
                                                           {17}{23456}
                                                           {1267}{35}{4}
                                                           {1357}{26}{4}
                                                           {147}{26}{35}
                                                           {156}{237}{4}
                                                           {17}{2356}{4}
                                                           {17}{246}{35}
                                                           {17}{26}{345}
                                                           {17}{26}{35}{4}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A000110, A007837, A035470, A038041, A070925, A275780, A306017, A306021, A319169, A326513, A326515, A326520, A326536.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Mean/@#&]],{n,0,8}]

a(12)-a(15) from Alois P. Heinz, Jul 12 2019

a(16)-a(26) from Christian Sievers, Aug 22 2024

A319612 Number of regular simple graphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 1, 7, 13, 171, 931, 45935, 1084413, 155862511, 10382960971, 6939278572095, 2203360500122299, 4186526756621772343, 3747344008241368443819, 35041787059691023579970847, 156277111373303386104606663421, 4142122641757598618318165240180095

Offset: 0

Gus Wiseman, Dec 17 2018

nonn

A graph is regular if all vertices have the same degree. The span of a graph is the union of its edges.

			The a(4) = 7 edge-sets:
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A002829, A005176, A110100, A116539, A295193, A306017, A319189, A319190.

a(n) = A295193(n) - 1.

a(16)-a(18) from Andrew Howroyd, Sep 02 2019

A301922 Regular triangle where T(n,k) is the number of unlabeled k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 23, 29, 4, 1, 1, 122, 2102, 150, 5, 1, 1, 888, 7011184, 7013164, 1037, 6, 1, 1, 11302, 1788775603336, 29281354507753848, 1788782615612, 12338, 7, 1, 1, 262322, 53304526022885280592, 234431745534048893449761040648512, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1

Offset: 1

Gus Wiseman, Jun 19 2018

			Triangle begins:
   1
   1   1
   1   2   1
   1   7   3   1
   1  23  29   4   1
The T(4,2) = 7 hypergraphs:
  {{1,2},{3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 1..91

Row sums are A301481. Second column is A002494.

Cf. A003465, A055621, A298422, A298426, A299471, A301481, A301920, A306017-A306021, A309858.

g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
     [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
     /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
     /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
    `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
     /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
A:= proc(n, k) A(n, k):= `if`(k>n-k, A(n, n-k), b(n$2, [], k)) end:
T:= (n, k)-> A(n, k)-A(n-1, k):
seq(seq(T(n, k), k=1..n), n=1..9);  # Alois P. Heinz, Aug 21 2019

permcount(v)={my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L,k); while(#L0, u=vecsort(apply(f, u)); d=lex(u,v)); !d}
Q(n,k,perm)={my(t=0); forsubset([n,k], v, t += can(Vec(v), t->perm[t])); t}
U(n,k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n,k,rep(p))); s/n!}
for(n=1, 10, for(k=1, n, print1(U(n,k)-U(n-1,k), ", ")); print) \\ Andrew Howroyd, Aug 10 2019

T(n,k) = A309858(n,k) - A309858(n-1,k). - Alois P. Heinz, Aug 21 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 09 2019

A322451 Number of unlabeled 3-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 29, 2102, 7011184, 1788775603336, 53304526022885280592, 366299663378889804782337225824, 1171638318502622784366970315264281830913536, 3517726593606524901243694560022510194223171115509135178240

Offset: 0

Gus Wiseman, Dec 09 2018

nonn

3-uniform means that every edge consists of 3 vertices. - Brendan McKay, Sep 03 2023

			Non-isomorphic representatives of the a(5) = 29 hypergraphs:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{123}{145}{245}{345}}
  {{124}{135}{245}{345}}
  {{125}{135}{245}{345}}
  {{134}{235}{245}{345}}
  {{145}{235}{245}{345}}
  {{123}{124}{135}{245}{345}}
  {{123}{145}{235}{245}{345}}
  {{124}{134}{235}{245}{345}}
  {{134}{145}{235}{245}{345}}
  {{135}{145}{235}{245}{345}}
  {{145}{234}{235}{245}{345}}
  {{123}{124}{134}{235}{245}{345}}
  {{123}{134}{145}{235}{245}{345}}
  {{123}{145}{234}{235}{245}{345}}
  {{124}{135}{145}{235}{245}{345}}
  {{125}{135}{145}{235}{245}{345}}
  {{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{235}{245}{345}}
  {{124}{135}{145}{234}{235}{245}{345}}
  {{125}{135}{145}{234}{235}{245}{345}}
  {{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{234}{235}{245}{345}}
  {{125}{134}{135}{145}{234}{235}{245}{345}}
  {{124}{125}{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}

Andrew Howroyd, Table of n, a(n) for n = 0..25

First differences of A000665.

Cf. A006126, A006129, A038041, A299471, A301922, A302374, A302394, A306017, A306021.

a(12) from Andrew Howroyd, Dec 15 2018

Name corrected by Brendan McKay, Sep 03 2023

cp's OEIS Frontend

A320325 Numbers whose product of prime indices is a perfect power.

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A000665 Number of 3-uniform hypergraphs on n unlabeled nodes, or equivalently number of relations with 3 arguments on n nodes.

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A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

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A319190 Number of regular hypergraphs spanning n vertices.

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A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

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A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

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A326512 Number of set partitions of {1..n} where every block has the same average.

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A319612 Number of regular simple graphs spanning n vertices.

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