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

1,2

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

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

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 *)

Crossrefs

Cf. A215366, A258116.

Sequence in context: A118667 A353357 A352140 * A034017 A034021 A216516

Adjacent sequences: A258114 A258115 A258116 * A258118 A258119 A258120

Keyword

nonn

Author

Emeric Deutsch, May 20 2015

Extensions

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

Status

approved