cp's OEIS Frontend

A328512 Number of distinct connected components of the multiset of multisets with MM-number n.

%I A328512 #10 Jan 28 2025 16:54:58
%S A328512 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 A328512 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 A328512 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 A328512 Number of distinct connected components of the multiset of multisets with MM-number n.
%C A328512 For n > 1, the first appearance of n is 2 * A080696(n - 1).
%C A328512 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 A328512 Antti Karttunen, <a href="/A328512/b328512.txt">Table of n, a(n) for n = 1..20000</a>
%H A328512 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A328512 If n is even, a(n) = A305079(n) - A007814(n) + 1; otherwise, a(n) = A305079(n).
%e A328512 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 A328512 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 A328512 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A328512 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 A328512 Table[Length[Union[zsm[primeMS[n]]]],{n,100}]
%o A328512 (PARI)
%o A328512 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 A328512 A007814(n) = valuation(n, 2);
%o A328512 A000265(n) = (n/2^A007814(n));
%o A328512 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 A328512 Positions of 0's and 1's are A305078 together with all powers of 2.
%Y A328512 Connected numbers are A305078.
%Y A328512 Connected components are A305079.
%Y A328512 Cf. A007814, A056239, A112798, A286518, A302242, A304714, A304716, A322389, A327076, A328513.
%K A328512 nonn
%O A328512 1,6
%A A328512 _Gus Wiseman_, Oct 20 2019
%E A328512 Data section extended to a(105) by _Antti Karttunen_, Jan 28 2025