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A305501 Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n. 3
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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.

This is intended to be a cleaner form of A305079, where the treatment of empty multisets is arbitrary.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n =  1..100000

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

EXAMPLE

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.

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[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

Table[Length[zsm[primeMS[n]+1]], {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); };

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

CROSSREFS

Cf. A001221, A048143, A056239, A112798, A275024, A286518, A302242, A303837, A304118, A304714, A304716, A305078, A305079, A305504.

Sequence in context: A326775 A317240 A326620 * A184170 A025919 A095684

Adjacent sequences:  A305498 A305499 A305500 * A305502 A305503 A305504

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 03 2018

EXTENSIONS

More terms from Antti Karttunen, Nov 09 2018

STATUS

approved

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Last modified July 26 01:49 EDT 2021. Contains 346294 sequences. (Running on oeis4.)