A325038 - OEIS

%I #5 Mar 26 2019 21:06:04

%S 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 60,62,64,68,72,74,76,80,82,86,88,92,94,96,104,106,112,116,118,120,

%U 122,124,128,134,136,142,144,146,148,152,158,160,164,166,168,172

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

%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 less than their sum of prime indices (A056239).

%C The enumeration of these partitions by sum is given by A096276 shifted once to the right.

%F A003963(a(n)) < A056239(a(n)).

%F a(n) = 2 * A325044(n).

%e The sequence of terms together with their prime indices begins:

%e    4: {1,1}

%e    6: {1,2}

%e    8: {1,1,1}

%e   10: {1,3}

%e   12: {1,1,2}

%e   14: {1,4}

%e   16: {1,1,1,1}

%e   18: {1,2,2}

%e   20: {1,1,3}

%e   22: {1,5}

%e   24: {1,1,1,2}

%e   26: {1,6}

%e   28: {1,1,4}

%e   32: {1,1,1,1,1}

%e   34: {1,7}

%e   36: {1,1,2,2}

%e   38: {1,8}

%e   40: {1,1,1,3}

%e   44: {1,1,5}

%e   46: {1,9}

%e   48: {1,1,1,1,2}

%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, A325037, A325041, A325042, A325044.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 25 2019