A325043 - OEIS

%I #9 Jun 28 2020 02:25:36

%S 18,60,168,216,400,528,1248,2240,2880,3264,7296,14080,17664,25088,

%T 32256,41472,44544,66560,95232,153600,227328,315392,348160,405504,

%U 503808,1056768,1556480,2310144,2981888,3833856,5210112,6881280,7536640,7929856,8847360,11599872

%N Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer 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 with at least three prime factors (counted with multiplicity) whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

%F a(n) = 2 * A301988(n).

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

%e      18: {1,2,2}

%e      60: {1,1,2,3}

%e     168: {1,1,1,2,4}

%e     216: {1,1,1,2,2,2}

%e     400: {1,1,1,1,3,3}

%e     528: {1,1,1,1,2,5}

%e    1248: {1,1,1,1,1,2,6}

%e    2240: {1,1,1,1,1,1,3,4}

%e    2880: {1,1,1,1,1,1,2,2,3}

%e    3264: {1,1,1,1,1,1,2,7}

%e    7296: {1,1,1,1,1,1,1,2,8}

%e   14080: {1,1,1,1,1,1,1,1,3,5}

%e   17664: {1,1,1,1,1,1,1,1,2,9}

%e   25088: {1,1,1,1,1,1,1,1,1,4,4}

%e   32256: {1,1,1,1,1,1,1,1,1,2,2,4}

%e   41472: {1,1,1,1,1,1,1,1,1,2,2,2,2}

%e   44544: {1,1,1,1,1,1,1,1,1,2,10}

%e   66560: {1,1,1,1,1,1,1,1,1,1,3,6}

%e   95232: {1,1,1,1,1,1,1,1,1,1,2,11}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[10000],And[PrimeOmega[#]>2,Times@@primeMS[#]==Total[primeMS[#]]-1]&]

%Y Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

%Y Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042, A325044.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 25 2019

%E More terms from _Jinyuan Wang_, Jun 27 2020