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A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

%I A325044 #4 Mar 26 2019 21:06:55
%S A325044 2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,22,23,24,26,28,29,30,
%T A325044 31,32,34,36,37,38,40,41,43,44,46,47,48,52,53,56,58,59,60,61,62,64,67,
%U A325044 68,71,72,73,74,76,79,80,82,83,84,86,88,89,92,94,96,97,101
%N A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.
%C A325044 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than or equal to their sum of prime indices (A056239).
%C A325044 The enumeration of these partitions by sum is given by A096276.
%F A325044 A003963(a(n)) <= A056239(a(n)).
%F A325044 a(n) = A325038(n)/2.
%F A325044 Union of A301987 and A325038.
%e A325044 The sequence of terms together with their prime indices begins:
%e A325044    2: {1}
%e A325044    3: {2}
%e A325044    4: {1,1}
%e A325044    5: {3}
%e A325044    6: {1,2}
%e A325044    7: {4}
%e A325044    8: {1,1,1}
%e A325044    9: {2,2}
%e A325044   10: {1,3}
%e A325044   11: {5}
%e A325044   12: {1,1,2}
%e A325044   13: {6}
%e A325044   14: {1,4}
%e A325044   16: {1,1,1,1}
%e A325044   17: {7}
%e A325044   18: {1,2,2}
%e A325044   19: {8}
%e A325044   20: {1,1,3}
%e A325044   22: {1,5}
%e A325044   23: {9}
%e A325044   24: {1,1,1,2}
%t A325044 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325044 Select[Range[100],Times@@primeMS[#]<=Plus@@primeMS[#]&]
%Y A325044 Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
%Y A325044 Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042.
%K A325044 nonn
%O A325044 1,1
%A A325044 _Gus Wiseman_, Mar 25 2019