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A305079
Number of connected components of the integer partition with Heinz number n.
63
0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 5, 2, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 4, 1, 2, 1, 6, 1, 3, 1, 3, 2, 3, 1, 4, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 1, 3, 2, 2, 1
OFFSET
1,4
COMMENTS
First differs from |A305052(n)| at a(169) = 1, A305052(169) = 0.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite multiset 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. If S is the integer partition with Heinz number n, a(n) is the number of connected components of G(S).
FORMULA
For all n, k > 0, we have a(2^n * k) = n + a(k).
For all x, y > 0, we have a(x * y) <= a(x) + a(y).
For x, y > 0 strongly coprime, we have a(x * y) = a(x) + a(y). Strongly coprime means every prime index of x is coprime to every prime index of y, where a prime index of n is a number m such that prime(m) divides n.
a(n) = A305501(A064989(n)) + A007814(n). - Antti Karttunen, Nov 10 2018
EXAMPLE
The a(315) = 2 connected components of {2,2,3,4} are {{3},{2,2,4}}.
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[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));
A305079(n) = if(!(n%2), A007814(n)+A305079(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, Nov 10 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 24 2018
EXTENSIONS
Terms and Mathematica program corrected by Gus Wiseman, Nov 10 2018
STATUS
approved