OFFSET
1,2
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, the Heinz number of the partition [1, 1, 2, 4, 10] is 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
More terms are obtained if one replaces the 350 in the Maple program by a larger number.
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
EXAMPLE
170 is in the sequence because it is the Heinz number of the partition [1,3,7]; indeed, (1st prime)*(3rd prime)*(7th prime) = 2*5*17 = 170.
MAPLE
with(numtheory): B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: DO := {}: for q to 350 do if `and`(nops(B(q)) = nops(convert(B(q), set)), map(type, convert(B(q), set), odd) = {true}) then DO := `union`(DO, {q}) else end if end do: DO;
# second Maple program:
a:= proc(n) option remember; local k;
for k from 1+`if`(n=1, 0, a(n-1)) do
if not false in map(i-> i[2]=1 and numtheory
[pi](i[1])::odd, ifactors(k)[2]) then break fi
od; k
end:
seq(a(n), n=1..100); # Alois P. Heinz, May 10 2016
MATHEMATICA
a[n_] := a[n] = Module[{k}, For[k = 1 + If[n == 1, 0, a[n-1]], True, k++, If[AllTrue[FactorInteger[k], #[[2]] == 1 && OddQ[PrimePi[#[[1]]]]&], Break[]]]; k]; Join[{1}, Array[a, 100]] (* Jean-François Alcover, Dec 10 2016 after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 20 2015
EXTENSIONS
a(1)=1 inserted by Alois P. Heinz, May 10 2016
STATUS
approved