A325037 - OEIS

%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