|
| |
|
|
A257992
|
|
Number of even parts in the partition having Heinz number n.
|
|
61
|
|
|
|
0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 2, 0, 0, 1, 0, 1, 3, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 1, 2, 0, 0, 2, 1, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 3, 0, 1, 2, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 0, 4, 0, 0, 2, 0, 1, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,9
|
|
|
COMMENTS
|
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
|
|
|
REFERENCES
|
G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
M. Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(18) = 2 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having 2 even parts.
|
|
|
MAPLE
|
with(numtheory): a := proc (n) local B, ct, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for q to nops(B(n)) do if `mod`(B(n)[q], 2) = 0 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 135);
# second Maple program:
a:= n-> add(`if`(numtheory[pi](i[1])::even, i[2], 0), i=ifactors(n)[2]):
|
|
|
MATHEMATICA
|
a[n_] := Sum[If[PrimePi[i[[1]]] // EvenQ, i[[2]], 0], {i, FactorInteger[n]} ]; a[1] = 0; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Dec 10 2016 after Alois P. Heinz *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
