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.
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 18 2015
STATUS
approved