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

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, 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, 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

Offset

1,6

Comments

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

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}}.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to prime indices in the factorization of n.

Formula

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

Example

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.
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.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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]]]]]]]]];
Table[Length[Union[zsm[primeMS[n]]]], {n, 100}]

Prog

(PARI)
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); };
A007814(n) = valuation(n, 2);
A000265(n) = (n/2^A007814(n));
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

Crossrefs

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

Connected numbers are A305078.

Connected components are A305079.

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

Sequence in context: A125029 A339839 A062893 * A302041 A302031 A371883

Adjacent sequences: A328509 A328510 A328511 * A328513 A328514 A328515

Keyword

nonn

Author

Gus Wiseman, Oct 20 2019

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

Status

approved