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

Author

Emeric Deutsch, May 18 2015

Keywords

• nonn

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.

Examples

			a(12) = 2 because the partition having Heinz number 12 = 2*2*3 is [1,1,2], having 2 odd parts.
		

References

• George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.
• Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Links

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

Crossrefs

Cf. A001222, A195017, A215366, A257992.

Programs

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 *)

Formula

From Amiram Eldar, Jun 17 2024: (Start)

Totally additive with a(p) = 1 if primepi(p) is odd, and 0 otherwise.

a(n) = A257992(n) + A195017(n). (End)