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

%I A258117 #17 Feb 14 2020 16:38:54
%S A258117 1,3,7,13,19,21,29,37,39,43,53,57,61,71,79,87,89,91,101,107,111,113,
%T A258117 129,131,133,139,151,159,163,173,181,183,193,199,203,213,223,229,237,
%U A258117 239,247,251,259,263,267,271,273,281,293,301,303,311,317,321,337,339,349
%N A258117 The Heinz numbers in increasing order of the partitions into distinct even parts.
%C A258117 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.
%C A258117 In the Maple program the subprogram B yields the partition with Heinz number n.
%C A258117 More terms are obtained if one replaces the 350 in the Maple program by a larger number.
%D A258117 G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976.
%D A258117 G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
%H A258117 Alois P. Heinz, <a href="/A258117/b258117.txt">Table of n, a(n) for n = 1..10000</a>
%e A258117 213 is in the sequence because it is the Heinz number of the partition [2,20]; indeed, (2nd prime)*(20th prime) = 3*71 = 213.
%p A258117 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;
%p A258117 # second Maple program:
%p A258117 a:= proc(n) option remember; local k;
%p A258117       for k from 1+`if`(n=1, 0, a(n-1)) do
%p A258117         if not false in map(i-> i[2]=1 and numtheory
%p A258117         [pi](i[1])::even, ifactors(k)[2]) then break fi
%p A258117       od; k
%p A258117     end:
%p A258117 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 10 2016
%t A258117 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_ *)
%Y A258117 Cf. A215366, A258116.
%K A258117 nonn
%O A258117 1,2
%A A258117 _Emeric Deutsch_, May 20 2015
%E A258117 a(1)=1 inserted by _Alois P. Heinz_, May 10 2016