A076078 -id:A076078 - cp's OEIS frontend

A000371 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)2^(2^k).

2, 2, 10, 218, 64594, 4294642034, 18446744047940725978, 340282366920938463334247399005993378250, 115792089237316195423570985008687907850547725730273056332267095982282337798562

Offset: 0

N. J. A. Sloane

Inverse binomial transform of A001146.
Number of nondegenerate Boolean functions of n variables.
Twice the number of covers of an n-set S (A003465). That is, the number of subsets of the power set of S whose union is S. [corrected by Manfred Boergens, May 02 2024]
From David P. Moulton, Nov 11 2010: (Start)
To see why the formula in the definition gives the number of covers of an n-set we use inclusion-exclusion.
The set S has n elements and T, the power set of S, has 2^n elements.
Let U be the power set of T; we want to know how many elements of U have union S.
For any element i of S, let U_i be the subset of U whose unions do not contain i, so we want to compute the size of the complement of the union of the U_i s.
Write U_I for the union of U_i for i in I. Then U_I consists of all subsets of T whose union is disjoint from I, so it consists of all subsets of the power set of S - I. The power set of S - I has 2^(n - #I) elements, so U_I has size 2^2^(n - #I).
Then the basic inclusion-exclusion formula says that our answer is
#(U - union_{i in S} U_i) = Sum_{I subseteq S} (-1)^#I #U_I = Sum_{j=0..n} (-1)^j Sum_{#I = j} #U_I = Sum_{j=0..n} (-1)^j binomial(n,j)*2^2^(n-j), as required.
(End)
Here is Comtet's proof: Let P'(S) be the power set of nonempty subsets of S. Then |P'(P'(S))| = 2^(2^n-1)-1 = Sum_k binomial(n,k)*a(k). Apply the inverse binomial transform to get a(n) = Sum_k (-1)^k*binomial(n,k)*2^(2^(n-k)-1). - N. J. A. Sloane, May 19 2011
For disjoint subsets of the power set see A186021. For disjoint nonempty subsets of the power set see A000110. - Manfred Boergens, May 02 2024 and Apr 09 2025

			Let n = 2, S = {a,b}, and P = {0,a,b,ab}. There are ten subsets of P whose union is S: {ab}, {a,b}, {a,ab}, {b,ab}, {a,b,ab}, and the empty set together with the same five. - _Marc LeBrun_, Nov 10 2010

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 165.
M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 170.
S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
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).
C. G. Wagner, Covers of finite sets, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 515-520.

R. Baumann and H. Strass, On the number of bipolar Boolean functions, 2014, preprint.
R. Baumann and H. Strass, On the number of bipolar Boolean functions, Journal of Logic and Computation, 27(8) (2017), 2431-2449.
Thomas M. A. Fink, Regulatory motifs: structural and functional building blocks of genetic computation, arXiv:2208.14996 [q-bio.MN], 2022.
Goto, Eiichi, and Hidetosi Takahasi, Some Theorems Useful in Threshold Logic for Enumerating Boolean Functions, in Proceedings International Federation for Information Processing (IFIP) Congress, 1962, pp. 747-752. [Annotated scans of certain pages]
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
M. Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.
A. J. Macula, Covers of a finite set, Math. Mag., 67 (1994), 141-144.
S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971 [Annotated scans of a few pages]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Cover
Index entries for sequences related to Boolean functions

Equals twice A003465.
Row sums of A163353.
Diagonal of A381683.
Cf. A001146, A002110, A076078, A055154.
Cf. A000110, A186021.

[&+[(-1)^(n-k)*Binomial(n, k)*2^(2^k): k in [0..n]]: n in [0..10]]; // Vincenzo Librandi, Dec 28 2015

f:=n->add((-1)^(n-k)*binomial(n,k)*2^(2^k),k=0..n);

Table[Sum[(-1)^(n-k) Binomial[n,k]2^(2^k),{k,0,n}],{n,0,10}] (* Harvey P. Dale, Oct 17 2011 *)

a(n)=sum(k=0,n,(-1)^(n-k)*binomial(n,k)<<(2^k)) \\ Charles R Greathouse IV, Jan 02 2012

a(n) = sum(k=0, n, (-1)^k*n!/k!/(n-k)!*2^(2^(n-k))); \\ Altug Alkan, Dec 29 2015

The coefficient of x^k in the polynomial p_n(x) = Sum_{j=0..n} (-1)^j binomial(n,j) * (x+1)^2^(n-j) gives the number of covers of a set of size n where the covers have k elements.  Also, there is a recurrence: f_n(k) = k, if n = 0, and f_n(k) = f_{n-1}(k^2) - f_{n-1}(k), if n > 0, that gives a(n) = f_n(2) and p_n(x) = f_n(x+1). - David W. Wilson, Nov 11 2010
E.g.f.: Sum(exp((2^n-1)*x)*log(2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 30 2004
For n > 0, a(n) = A076078(A002110(n)). - Matthew Vandermast, Nov 14 2010
a(n) ~ 2^(2^n). - Charles R Greathouse IV, Jan 02 2012
a(n) = 2*A003465(n). - Maurizio De Leo, Feb 27 2015

Since this sequence arises in several different contexts, I replaced the old definition with an explicit formula. - N. J. A. Sloane, Nov 23 2010

A285573 Number of finite nonempty sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 9, 2, 5, 5, 5, 2, 9, 2, 9, 5, 5, 2, 14, 3, 5, 4, 9, 2, 19, 2, 6, 5, 5, 5, 19, 2, 5, 5, 14, 2, 19, 2, 9, 9, 5, 2, 20, 3, 9, 5, 9, 2, 14, 5, 14, 5, 5, 2, 49, 2, 5, 9, 7, 5, 19, 2, 9, 5, 19, 2, 34, 2, 5, 9, 9, 5, 19, 2, 20, 5, 5, 2, 49, 5, 5, 5, 14, 2, 49, 5, 9, 5, 5, 5, 27, 2, 9, 9, 19

Offset: 1

Gus Wiseman, Apr 21 2017

nonn

From Robert Israel, Apr 21 2017: (Start)
If n = p^k for prime p, a(n) = k+1.
If n = p^j*q^k for distinct primes p,q, a(n) = binomial(j+k+2,j+1)-1. (End)

			The a(12)=9 sets are: {1}, {2}, {3}, {4}, {6}, {12}, {2,3}, {3,4}, {4,6}.

Cf. A006126, A048143, A076078, A076413, A198085, A285572.

g:= proc(S) local x, Sx; option remember;
   if nops(S) = 0 then return {{}} fi;
   x:= S[1];
   Sx:= subsop(1=NULL,S);
   procname(Sx) union map(t -> t union {x}, procname(remove(s -> s mod x = 0 or x mod s = 0, Sx)))
end proc:
f:= proc(n) local F,D;
  F:= ifactors(n)[2];
  D:= numtheory:-divisors(mul(ithprime(i)^F[i,2],i=1..nops(F)));
  nops(g(D)) - 1;
end proc:
map(f, [$1..100]); # Robert Israel, Apr 21 2017

nn=50;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Rest[stableSets[Divisors[n],Divisible]]],{n,1,nn}]

A304714 Number of connected strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 5, 10, 6, 12, 12, 13, 14, 21, 17, 23, 26, 30, 31, 46, 38, 51, 55, 61, 70, 87, 85, 102, 116, 128, 138, 171, 169, 204, 225, 245, 272, 319, 334, 383, 429, 464, 515, 593, 629, 715, 790, 861, 950, 1082

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(19) = 6 strict integer partitions are (19), (9,6,4), (10,5,4), (10,6,3), (12,4,3), (8,6,3,2). Taking the normalized prime factors of each part (see A112798, A302242), we have the following connected multiset multisystems.
       (19): {{8}}
    (9,6,4): {{2,2},{1,2},{1,1}}
   (10,5,4): {{1,3},{3},{1,1}}
   (10,6,3): {{1,3},{1,2},{2}}
   (12,4,3): {{1,1,2},{1,1},{2}}
  (8,6,3,2): {{1,1,1},{1,2},{2},{1}}

The Heinz numbers of these partitions are given by A328513.

Cf. A000009, A003963, A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A302242.

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&]],{n,60}]

A286520 Number of finite connected sets of pairwise indivisible positive integers greater than one with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 5, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 17, 1, 1, 2, 1, 1, 5, 1, 2, 1, 5, 1, 9, 1, 1, 2, 2, 1, 5, 1, 4, 1, 1, 1, 17, 1, 1, 1

Offset: 2

Gus Wiseman, Jul 24 2017

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 that are not relatively prime. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

			The a(30)=5 sets are: {30}, {6,10}, {6,15}, {10,15}, {6,10,15}.

Cf. A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518.

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[Subsets[Rest[Divisors[n]]],And[!MemberQ[Tuples[#,2],{x_,y_}/;And[x

A286518 Number of finite connected sets of positive integers greater than one with least common multiple n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 2, 4, 1, 20, 1, 4, 4, 8, 1, 20, 1, 20, 4, 4, 1, 88, 2, 4, 4, 20, 1, 96, 1, 16, 4, 4, 4, 196, 1, 4, 4, 88, 1, 96, 1, 20, 20, 4, 1, 368, 2, 20, 4, 20, 1, 88, 4, 88, 4, 4, 1, 1824, 1, 4, 20, 32, 4, 96, 1, 20, 4, 96, 1, 1688, 1, 4, 20, 20, 4, 96, 1, 368, 8, 4, 1, 1824, 4, 4, 4, 88, 1, 1824, 4, 20

Offset: 1

Gus Wiseman, Jul 24 2017

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 that are not relatively prime. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Feb 17 2024

			The a(6)=4 sets are: {6}, {2,6}, {3,6}, {2,3,6}.

Antti Karttunen, Table of n, a(n) for n = 1..719
Index entries for sequences computed from exponents in factorization of n

Cf. A048143, A054921, A069626, A076078, A259936, A281116, A285572, A285573, A286520, A305193, A318670.

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[Subsets[Rest[Divisors[n]]],zsm[#]==={n}&]],{n,2,20}]

isconnected(facs) = { my(siz=length(facs)); if(1==siz,1,my(m=matrix(siz,siz,i,j,(gcd(facs[i],facs[j])!=1))^siz); for(n=1,siz,if(0==vecmin(m[n,]),return(0))); (1)); };
A286518aux(n, parts, from=1, ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==n && isconnected(ss), s++); for(i=from, k, newss = List(ss); listput(newss, parts[i]); s += A286518aux(n, parts, i+1, newss)); (s) };
A286518(n) = if(1==n, n, A286518aux(n, divisors(n))); \\ Antti Karttunen, Feb 17 2024

From Antti Karttunen, Feb 17 2024: (Start)
a(n) <= A069626(n).
It seems that a(n) >= A318670(n), for all n > 1.
(End)

Term a(1)=1 prepended and more terms added by Antti Karttunen, Feb 17 2024

A303837 Number of z-trees with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 10, 1, 1, 2, 1, 1, 4, 1, 2, 1, 4, 1, 6, 1, 1, 2, 2, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. Then a z-tree is a finite connected set of pairwise indivisible positive integers greater than 1 with clutter density -1.
This is a generalization to multiset systems of the usual definition of hypertree (viz. connected hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A030019(k).

			The a(72) = 6 z-trees together with the corresponding multiset systems (see A112798, A302242) are the following.
      (72): {{1,1,1,2,2}}
    (8,18): {{1,1,1},{1,2,2}}
    (8,36): {{1,1,1},{1,1,2,2}}
    (9,24): {{2,2},{1,1,1,2}}
   (6,8,9): {{1,2},{1,1,1},{2,2}}
  (8,9,12): {{1,1,1},{2,2},{1,1,2}}
The a(60) = 10 z-trees together with the corresponding multiset systems are the following.
       (60): {{1,1,2,3}}
     (4,30): {{1,1},{1,2,3}}
     (6,20): {{1,2},{1,1,3}}
    (10,12): {{1,3},{1,1,2}}
    (12,15): {{1,1,2},{2,3}}
    (12,20): {{1,1,2},{1,1,3}}
    (15,20): {{2,3},{1,1,3}}
   (4,6,10): {{1,1},{1,2},{1,3}}
   (4,6,15): {{1,1},{1,2},{2,3}}
  (4,10,15): {{1,1},{1,3},{2,3}}

Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A006126, A030019, A048143, A076078, A112798, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303838, A304118.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zensity[#]==-1,zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,2,50}]

A304118 Number of z-blobs with least common multiple n > 1.

Original entry on oeis.org

0, 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, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-blob is a finite connected set S of pairwise indivisible positive integers greater than 1 such that no cap of S with at least two edges has clutter density -1.

If n is squarefree with k prime factors, then a(n) = A275307(k).

			The a(60) = 7 z-blobs together with the corresponding multiset systems (see A112798, A302242) are the following.
        (60): {{1,1,2,3}}
     (12,30): {{1,1,2},{1,2,3}}
     (20,30): {{1,1,3},{1,2,3}}
   (6,15,20): {{1,2},{2,3},{1,1,3}}
  (10,12,15): {{1,3},{1,1,2},{2,3}}
  (12,15,20): {{1,1,2},{2,3},{1,1,3}}
  (12,20,30): {{1,1,2},{1,1,3},{1,2,3}}
The a(120) = 14 z-blobs together with the corresponding multiset systems are the following.
       (120): {{1,1,1,2,3}}
     (24,30): {{1,1,1,2},{1,2,3}}
     (24,60): {{1,1,1,2},{1,1,2,3}}
     (30,40): {{1,2,3},{1,1,1,3}}
     (40,60): {{1,1,1,3},{1,1,2,3}}
   (6,15,40): {{1,2},{2,3},{1,1,1,3}}
  (10,15,24): {{1,3},{2,3},{1,1,1,2}}
  (12,15,40): {{1,1,2},{2,3},{1,1,1,3}}
  (12,30,40): {{1,1,2},{1,2,3},{1,1,1,3}}
  (15,20,24): {{2,3},{1,1,3},{1,1,1,2}}
  (15,24,40): {{2,3},{1,1,1,2},{1,1,1,3}}
  (20,24,30): {{1,1,3},{1,1,1,2},{1,2,3}}
  (24,30,40): {{1,1,1,2},{1,2,3},{1,1,1,3}}
  (24,40,60): {{1,1,1,2},{1,1,1,3},{1,1,2,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A303838.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
zlobQ[s_]:=Apply[And,Composition[Not,zreeQ]/@Apply[LCM,zptns[s],{2}]];
zswell[s_]:=Union[LCM@@@Select[Subsets[s],Length[zsm[#]]==1&]];
zkernels[s_]:=Table[Select[s,Divisible[w,#]&],{w,zswell[s]}];
zptns[s_]:=Select[stableSets[zkernels[s],Length[Intersection[#1,#2]]>0&],Union@@#==s&];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[If[n==1,0,Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={},zlobQ[#]]&]]],{n,100}]

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421

Offset: 0

Gus Wiseman, Mar 13 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.

Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 15 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A302697 Odd numbers whose prime indices are relatively prime. Heinz numbers of integer partitions with no 1's and with relatively prime parts.

Original entry on oeis.org

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 309, 315, 323, 327, 329

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

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

			Sequence of integer partitions with no 1's and with relatively prime parts begins:
015: (3,2)
033: (5,2)
035: (4,3)
045: (3,2,2)
051: (7,2)
055: (5,3)
069: (9,2)
075: (3,3,2)
077: (5,4)
085: (7,3)
093: (11,2)
095: (8,3)
099: (5,2,2)
105: (4,3,2)
119: (7,4)
123: (13,2)
135: (3,2,2,2)

Cf. A000837, A000961, A001222, A005117, A007359, A051424, A076078, A101268, A275024, A285572, A289509, A298748, A302568, A302569, A302696, A302698.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,200,2],GCD@@primeMS[#]===1&]

A324741 Number of subsets of {1...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 19, 30, 54, 96, 156, 248, 440, 688, 1120, 1864, 3664, 5856, 11232, 16896, 31296, 53952, 91008, 137472, 270528, 516720, 863088, 1710816, 3173856, 4836672, 9329472, 14897376, 29788128, 52256448, 88429248, 166037184, 331648704, 497685888, 829449600

Offset: 0

Gus Wiseman, Mar 15 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 a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {2,4}
                              {3,4}    {2,5}
                              {4,5}    {3,4}
                              {2,4,5}  {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {2,4,5}
                                       {3,4,6}
                                       {4,5,6}
An example for n = 20 is {5,6,7,9,10,12,14,15,16,19,20}, with prime indices:
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  19: {8}
  20: {1,1,3}
None of these prime indices {1,2,3,4,8} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The maximal case is A324743. The strict integer partition version is A324751. The integer partition version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324742.

Table[Length[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

cp's OEIS Frontend

A000371 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*2^(2^k).

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A285573 Number of finite nonempty sets of pairwise indivisible divisors of n.

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A304714 Number of connected strict integer partitions of n.

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A286520 Number of finite connected sets of pairwise indivisible positive integers greater than one with least common multiple n.

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A286518 Number of finite connected sets of positive integers greater than one with least common multiple n.

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A303837 Number of z-trees with least common multiple n > 1.

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A304118 Number of z-blobs with least common multiple n > 1.

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A324736 Number of subsets of {1...n} containing all prime indices of the elements.

A000371 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)2^(2^k).