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A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

%I A325037 #7 Mar 27 2019 12:28:16
%S A325037 1,15,21,25,27,33,35,39,42,45,49,50,51,54,55,57,63,65,66,69,70,75,77,
%T A325037 78,81,85,87,90,91,93,95,98,99,100,102,105,110,111,114,115,117,119,
%U A325037 121,123,125,126,129,130,132,133,135,138,140,141,143,145,147,150,153
%N A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.
%C A325037 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 greater than their sum of prime indices (A056239).
%C A325037 The enumeration of these partitions by sum is given by A114324.
%H A325037 Alois P. Heinz, <a href="/A325037/b325037.txt">Table of n, a(n) for n = 1..10000</a>
%F A325037 A003963(a(n)) > A056239(a(n)).
%e A325037 The sequence of terms together with their prime indices begins:
%e A325037    1: {}
%e A325037   15: {2,3}
%e A325037   21: {2,4}
%e A325037   25: {3,3}
%e A325037   27: {2,2,2}
%e A325037   33: {2,5}
%e A325037   35: {3,4}
%e A325037   39: {2,6}
%e A325037   42: {1,2,4}
%e A325037   45: {2,2,3}
%e A325037   49: {4,4}
%e A325037   50: {1,3,3}
%e A325037   51: {2,7}
%e A325037   54: {1,2,2,2}
%e A325037   55: {3,5}
%e A325037   57: {2,8}
%e A325037   63: {2,2,4}
%e A325037   65: {3,6}
%e A325037   66: {1,2,5}
%e A325037   69: {2,9}
%e A325037   70: {1,3,4}
%e A325037   75: {2,3,3}
%e A325037   77: {4,5}
%e A325037   78: {1,2,6}
%e A325037   81: {2,2,2,2}
%p A325037 q:= n-> (l-> mul(i, i=l)>add(i, i=l))(map(i->
%p A325037     numtheory[pi](i[1])$i[2], ifactors(n)[2])):
%p A325037 select(q, [$1..200])[];  # _Alois P. Heinz_, Mar 27 2019
%t A325037 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325037 Select[Range[100],Times@@primeMS[#]>Plus@@primeMS[#]&]
%Y A325037 Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
%Y A325037 Cf. A325032, A325033, A325036, A325038, A325041, A325042, A325044.
%K A325037 nonn
%O A325037 1,2
%A A325037 _Gus Wiseman_, Mar 25 2019