A258117 The Heinz numbers in increasing order of the partitions into distinct even parts.

1, 3, 7, 13, 19, 21, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 193, 199, 203, 213, 223, 229, 237, 239, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317, 321, 337, 339, 349

Offset: 1

Author

Emeric Deutsch, May 20 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, 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.

Examples

			213 is in the sequence because it is the Heinz number of the partition [2,20]; indeed, (2nd prime)*(20th prime) = 3*71 = 213.
		

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

Crossrefs

Cf. A215366, A258116.

Programs

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: DE := {}: for q to 350 do if `and`(nops(B(q)) = nops(convert(B(q), set)), map(type, convert(B(q), set), even) = {true}) then DE := `union`(DE, {q}) else  end if end do: DE;
# 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])::even, 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 && EvenQ[PrimePi[#[[1]]]]&], Break[]]]; k]; Array[a, 100] (* Jean-François Alcover, Dec 12 2016 after Alois P. Heinz *)

Extensions

a(1)=1 inserted by Alois P. Heinz, May 10 2016