A257992 Number of even parts in the partition having Heinz number n.

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

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

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

Formula

From Amiram Eldar, Jun 17 2024: (Start)

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

a(n) = A257991(n) - A195017(n). (End)

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]):
seq(a(n), n=1..120);  # Alois P. Heinz, May 09 2016

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

Cf. A001222, A195017, A215366, A257991.

Sequence in context: A257217 A184154 A284441 * A060952 A297155 A037844

Adjacent sequences: A257989 A257990 A257991 * A257993 A257994 A257995

Keyword

nonn

Author

Emeric Deutsch, May 18 2015

Status

approved