A257991 Number of odd parts in the partition having Heinz number n.

0, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 2, 0, 1, 1, 4, 1, 1, 0, 3, 0, 2, 1, 3, 2, 1, 0, 2, 0, 2, 1, 5, 1, 2, 1, 2, 0, 1, 0, 4, 1, 1, 0, 3, 1, 2, 1, 4, 0, 3, 1, 2, 0, 1, 2, 3, 0, 1, 1, 3, 0, 2, 0, 6, 1, 2, 1, 3, 1, 2, 0, 3, 1, 1, 2, 2, 1, 1, 0, 5, 0, 2, 1, 2, 2, 1, 0, 4

Offset

1,4

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

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Example

a(12) = 2 because the partition having Heinz number 12 = 2*2*3 is [1,1,2], having 2 odd 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) = 1 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])::odd, i[2], 0), i=ifactors(n)[2]):
seq(a(n), n=1..120);  # Alois P. Heinz, May 09 2016

Mathematica

a[n_] := Sum[If[PrimePi[i[[1]]] // OddQ, i[[2]], 0], {i, FactorInteger[n]} ]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Dec 10 2016, after Alois P. Heinz *)

Crossrefs

Cf. A001222, A215366, A257992.

Sequence in context: A113949 A318808 A349935 * A343029 A343037 A152434

Adjacent sequences: A257988 A257989 A257990 * A257992 A257993 A257994

Keyword

nonn

Author

Emeric Deutsch, May 18 2015

Status

approved