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%I #8 Aug 09 2019 12:43:43
%S 2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,30,31,32,37,41,43,47,49,53,
%T 59,61,64,67,71,73,79,81,83,84,89,97,101,103,107,109,113,121,125,127,
%U 128,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197
%N Heinz numbers of integer partitions whose length and maximum both divide their sum.
%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 A326843.
%H R. J. Mathar, <a href="/A326837/b326837.txt">Table of n, a(n) for n = 1..505</a>
%e The sequence of terms together with their prime indices begins:
%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 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}
%e 32: {1,1,1,1,1}
%e 37: {12}
%p isA326837 := proc(n)
%p psigsu := A056239(n) ;
%p psigma := A061395(n) ;
%p psigle := numtheory[bigomega](n) ;
%p if modp(psigsu,psigma) = 0 and modp(psigsu,psigle) = 0 then
%p true;
%p else
%p false;
%p end if;
%p end proc:
%p n := 1:
%p for i from 2 to 3000 do
%p if isA326837(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[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Max[y]]&&Divisible[Total[y],Length[y]]]&]
%Y The non-constant case is A326838.
%Y The strict case is A326851.
%Y Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jul 26 2019
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