OFFSET
1,9
COMMENTS
The Durfee square of a partition is the largest square that fits inside the Ferrers board of the partition.
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.
First appearance of k is a(prime(k)^k) = k. - Gus Wiseman, Apr 12 2019
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976.
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
FORMULA
For a partition (p_1 >= p_2 >= ... > = p_r) the side-length of its Durfee square is the largest i such that p_i >=i.
EXAMPLE
a(9)=2; indeed, 9 = 3*3 is the Heinz number of the partition [2,2] and, clearly its Durfee square has side-length =2.
MAPLE
with(numtheory): a := proc (p) local B, S, i: 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: S := {}: for i to nops(B(p)) do if i <= B(p)[nops(B(p))+1-i] then S := `union`(S, {i}) else end if end do: max(S) end proc: seq(a(n), n = 2 .. 146);
# second Maple program:
a:= proc(n) local l, t;
l:= sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`);
for t from nops(l) to 1 by -1 do if l[t]>=t then break fi od; t
end:
seq(a(n), n=1..120); # Alois P. Heinz, May 10 2016
MATHEMATICA
a[n_] := a[n] = Module[{l, t}, l = Reverse[Sort[Flatten[Table[PrimePi[ f[[1]] ], {f, FactorInteger[n]}, {f[[2]]}]]]]; For[t = Length[l], t >= 1, t--, If[l[[t]] >= t, Break[]]]; t]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Feb 17 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 18 2015
EXTENSIONS
a(1)=0 prepended by Alois P. Heinz, May 10 2016
STATUS
approved