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A258116 The Heinz numbers in increasing order of the partitions into distinct odd parts.

%I A258116 #18 Feb 14 2020 16:37:27
%S A258116 1,2,5,10,11,17,22,23,31,34,41,46,47,55,59,62,67,73,82,83,85,94,97,
%T A258116 103,109,110,115,118,127,134,137,146,149,155,157,166,167,170,179,187,
%U A258116 191,194,197,205,206,211,218,227,230,233,235,241,253,254,257,269,274,277,283,295,298,307,310,313,314,331,334,335,341,347
%N A258116 The Heinz numbers in increasing order of the partitions into distinct odd parts.
%C A258116 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 A258116 In the Maple program the subprogram B yields the partition with Heinz number n.
%C A258116 More terms are obtained if one replaces the 350 in the Maple program by a larger number.
%D A258116 G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976.
%D A258116 G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
%H A258116 Alois P. Heinz, <a href="/A258116/b258116.txt">Table of n, a(n) for n = 1..10000</a>
%e A258116 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.
%p A258116 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;
%p A258116 # second Maple program:
%p A258116 a:= proc(n) option remember; local k;
%p A258116       for k from 1+`if`(n=1, 0, a(n-1)) do
%p A258116         if not false in map(i-> i[2]=1 and numtheory
%p A258116         [pi](i[1])::odd, ifactors(k)[2]) then break fi
%p A258116       od; k
%p A258116     end:
%p A258116 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 10 2016
%t A258116 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_ *)
%Y A258116 Cf. A215366, A258117.
%K A258116 nonn
%O A258116 1,2
%A A258116 _Emeric Deutsch_, May 20 2015
%E A258116 a(1)=1 inserted by _Alois P. Heinz_, May 10 2016