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%I #25 Nov 09 2018 22:02:23
%S 0,1,1,1,1,2,1,1,1,1,1,2,1,2,2,1,1,2,1,1,2,1,1,2,1,2,1,2,1,2,1,1,1,1,
%T 2,2,1,2,2,1,1,3,1,1,2,1,1,2,1,1,2,2,1,2,1,2,1,2,1,2,1,1,2,1,2,1,1,1,
%U 2,2,1,2,1,2,2,2,2,3,1,1,1,1,1,3,1,2,2,1,1,2,2,1,1,1,2,2,1,2,1,1,1,2,1,2,3
%N Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C 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 greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
%C This is intended to be a cleaner form of A305079, where the treatment of empty multisets is arbitrary.
%H Antti Karttunen, <a href="/A305501/b305501.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A305501/a305501.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%e The "prime index plus 1" multiset of 7410 is {2,3,4,7,9}, with connected components {{2,4},{3,9},{7}}, so a(7410) = 3.
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t 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]]]]]]]]];
%t Table[Length[zsm[primeMS[n]+1]],{n,100}]
%o (PARI)
%o 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); };
%o A305501(n) = { my(cs = apply(p -> 1+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 09 2018
%Y Cf. A001221, A048143, A056239, A112798, A275024, A286518, A302242, A303837, A304118, A304714, A304716, A305078, A305079, A305504.
%K nonn
%O 1,6
%A _Gus Wiseman_, Jun 03 2018
%E More terms from _Antti Karttunen_, Nov 09 2018
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