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Number of distinct connected components of the multiset of multisets with MM-number n.
0

%I #10 Jan 28 2025 16:54:58

%S 0,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,1,2,1,2,1,2,1,2,1,3,1,1,2,2,

%T 2,2,1,2,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,2,2,1,2,1,3,1,2,1,1,1,3,1,2,

%U 2,3,1,2,1,2,2,2,2,2,1,2,1,2,1,2,2,2,1,2,1,3,1,2,2,2,2,2,1,2,2,2,1,3,1,2,2

%N Number of distinct connected components of the multiset of multisets with MM-number n.

%C For n > 1, the first appearance of n is 2 * A080696(n - 1).

%C 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}}.

%H Antti Karttunen, <a href="/A328512/b328512.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.

%F If n is even, a(n) = A305079(n) - A007814(n) + 1; otherwise, a(n) = A305079(n).

%e The multiset of multisets with MM-number 1508 is {{},{},{1,2},{1,3}}, which has the 3 connected components {{}}, {{}}, and {{1,2},{1,3}}, two of which are distinct, so a(1508) = 2.

%e The multiset of multisets with MM-number 12818 is {{},{1,2},{4},{1,3}}, which has the 3 connected components {{}}, {{1,2},{1,3}}, and {{4}}, so a(12818) = 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[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t Table[Length[Union[zsm[primeMS[n]]]],{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 A007814(n) = valuation(n, 2);

%o A000265(n) = (n/2^A007814(n));

%o A328512(n) = if(!(n%2), 1+A328512(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_, Jan 28 2025

%Y Positions of 0's and 1's are A305078 together with all powers of 2.

%Y Connected numbers are A305078.

%Y Connected components are A305079.

%Y Cf. A007814, A056239, A112798, A286518, A302242, A304714, A304716, A322389, A327076, A328513.

%K nonn,changed

%O 1,6

%A _Gus Wiseman_, Oct 20 2019

%E Data section extended to a(105) by _Antti Karttunen_, Jan 28 2025