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A326837 Heinz numbers of integer partitions whose length and maximum both divide their sum.

%I A326837 #8 Aug 09 2019 12:43:43
%S A326837 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 A326837 59,61,64,67,71,73,79,81,83,84,89,97,101,103,107,109,113,121,125,127,
%U A326837 128,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197
%N A326837 Heinz numbers of integer partitions whose length and maximum both divide their sum.
%C A326837 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A326837 The enumeration of these partitions by sum is given by A326843.
%H A326837 R. J. Mathar, <a href="/A326837/b326837.txt">Table of n, a(n) for n = 1..505</a>
%e A326837 The sequence of terms together with their prime indices begins:
%e A326837     2: {1}
%e A326837     3: {2}
%e A326837     4: {1,1}
%e A326837     5: {3}
%e A326837     7: {4}
%e A326837     8: {1,1,1}
%e A326837     9: {2,2}
%e A326837    11: {5}
%e A326837    13: {6}
%e A326837    16: {1,1,1,1}
%e A326837    17: {7}
%e A326837    19: {8}
%e A326837    23: {9}
%e A326837    25: {3,3}
%e A326837    27: {2,2,2}
%e A326837    29: {10}
%e A326837    30: {1,2,3}
%e A326837    31: {11}
%e A326837    32: {1,1,1,1,1}
%e A326837    37: {12}
%p A326837 isA326837 := proc(n)
%p A326837     psigsu := A056239(n) ;
%p A326837     psigma := A061395(n) ;
%p A326837     psigle := numtheory[bigomega](n) ;
%p A326837     if modp(psigsu,psigma) = 0 and modp(psigsu,psigle) = 0 then
%p A326837         true;
%p A326837     else
%p A326837         false;
%p A326837     end if;
%p A326837 end proc:
%p A326837 n := 1:
%p A326837 for i from 2 to 3000 do
%p A326837     if isA326837(i) then
%p A326837         printf("%d %d\n",n,i);
%p A326837         n := n+1 ;
%p A326837     end if;
%p A326837 end do: # _R. J. Mathar_, Aug 09 2019
%t A326837 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 A326837 The non-constant case is A326838.
%Y A326837 The strict case is A326851.
%Y A326837 Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848.
%K A326837 nonn
%O A326837 1,1
%A A326837 _Gus Wiseman_, Jul 26 2019