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A326841 Heinz numbers of integer partitions of m >= 0 using divisors of m.

%I A326841 #7 Aug 09 2019 12:15:31
%S A326841 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,30,31,32,36,37,40,41,
%T A326841 43,47,48,49,53,59,61,63,64,67,71,73,79,81,83,84,89,97,101,103,107,
%U A326841 108,109,112,113,121,125,127,128,131,137,139,144,149,151,157,163
%N A326841 Heinz numbers of integer partitions of m >= 0 using divisors of m.
%C A326841 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A326841 The enumeration of these partitions by sum is given by A018818.
%H A326841 R. J. Mathar, <a href="/A326841/b326841.txt">Table of n, a(n) for n = 1..543</a>
%e A326841 The sequence of terms together with their prime indices begins:
%e A326841     1: {}
%e A326841     2: {1}
%e A326841     3: {2}
%e A326841     4: {1,1}
%e A326841     5: {3}
%e A326841     7: {4}
%e A326841     8: {1,1,1}
%e A326841     9: {2,2}
%e A326841    11: {5}
%e A326841    12: {1,1,2}
%e A326841    13: {6}
%e A326841    16: {1,1,1,1}
%e A326841    17: {7}
%e A326841    19: {8}
%e A326841    23: {9}
%e A326841    25: {3,3}
%e A326841    27: {2,2,2}
%e A326841    29: {10}
%e A326841    30: {1,2,3}
%e A326841    31: {11}
%p A326841 isA326841 := proc(n)
%p A326841     local ifs,psigsu,p,psig ;
%p A326841     psigsu := A056239(n) ;
%p A326841     for ifs in ifactors(n)[2] do
%p A326841         p := op(1,ifs) ;
%p A326841         psig := numtheory[pi](p) ;
%p A326841         if modp(psigsu,psig) <> 0 then
%p A326841             return false;
%p A326841         end if;
%p A326841     end do:
%p A326841     true;
%p A326841 end proc:
%p A326841 for i from 1 to 3000 do
%p A326841     if isA326841(i) then
%p A326841         printf("%d %d\n",n,i);
%p A326841         n := n+1 ;
%p A326841     end if;
%p A326841 end do: # _R. J. Mathar_, Aug 09 2019
%t A326841 Select[Range[100],With[{y=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And@@IntegerQ/@(Total[y]/y)]&]
%Y A326841 The case where the length also divides m is A326847.
%Y A326841 Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.
%K A326841 nonn
%O A326841 1,2
%A A326841 _Gus Wiseman_, Jul 26 2019