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