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