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A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

%I A325038 #5 Mar 26 2019 21:06:04
%S A325038 4,6,8,10,12,14,16,18,20,22,24,26,28,32,34,36,38,40,44,46,48,52,56,58,
%T A325038 60,62,64,68,72,74,76,80,82,86,88,92,94,96,104,106,112,116,118,120,
%U A325038 122,124,128,134,136,142,144,146,148,152,158,160,164,166,168,172
%N A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.
%C A325038 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 their sum of prime indices (A056239).
%C A325038 The enumeration of these partitions by sum is given by A096276 shifted once to the right.
%F A325038 A003963(a(n)) < A056239(a(n)).
%F A325038 a(n) = 2 * A325044(n).
%e A325038 The sequence of terms together with their prime indices begins:
%e A325038    4: {1,1}
%e A325038    6: {1,2}
%e A325038    8: {1,1,1}
%e A325038   10: {1,3}
%e A325038   12: {1,1,2}
%e A325038   14: {1,4}
%e A325038   16: {1,1,1,1}
%e A325038   18: {1,2,2}
%e A325038   20: {1,1,3}
%e A325038   22: {1,5}
%e A325038   24: {1,1,1,2}
%e A325038   26: {1,6}
%e A325038   28: {1,1,4}
%e A325038   32: {1,1,1,1,1}
%e A325038   34: {1,7}
%e A325038   36: {1,1,2,2}
%e A325038   38: {1,8}
%e A325038   40: {1,1,1,3}
%e A325038   44: {1,1,5}
%e A325038   46: {1,9}
%e A325038   48: {1,1,1,1,2}
%t A325038 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325038 Select[Range[100],Times@@primeMS[#]<Plus@@primeMS[#]&]
%Y A325038 Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
%Y A325038 Cf. A325032, A325033, A325036, A325037, A325041, A325042, A325044.
%K A325038 nonn
%O A325038 1,1
%A A325038 _Gus Wiseman_, Mar 25 2019