A325043 Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer than their sum.

18, 60, 168, 216, 400, 528, 1248, 2240, 2880, 3264, 7296, 14080, 17664, 25088, 32256, 41472, 44544, 66560, 95232, 153600, 227328, 315392, 348160, 405504, 503808, 1056768, 1556480, 2310144, 2981888, 3833856, 5210112, 6881280, 7536640, 7929856, 8847360, 11599872

Offset: 1

Author

Gus Wiseman, Mar 25 2019

Keywords

• nonn

Comments

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).

Examples

			The sequence of terms together with their prime indices begins:
     18: {1,2,2}
     60: {1,1,2,3}
    168: {1,1,1,2,4}
    216: {1,1,1,2,2,2}
    400: {1,1,1,1,3,3}
    528: {1,1,1,1,2,5}
   1248: {1,1,1,1,1,2,6}
   2240: {1,1,1,1,1,1,3,4}
   2880: {1,1,1,1,1,1,2,2,3}
   3264: {1,1,1,1,1,1,2,7}
   7296: {1,1,1,1,1,1,1,2,8}
  14080: {1,1,1,1,1,1,1,1,3,5}
  17664: {1,1,1,1,1,1,1,1,2,9}
  25088: {1,1,1,1,1,1,1,1,1,4,4}
  32256: {1,1,1,1,1,1,1,1,1,2,2,4}
  41472: {1,1,1,1,1,1,1,1,1,2,2,2,2}
  44544: {1,1,1,1,1,1,1,1,1,2,10}
  66560: {1,1,1,1,1,1,1,1,1,1,3,6}
  95232: {1,1,1,1,1,1,1,1,1,1,2,11}
		

Crossrefs

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],And[PrimeOmega[#]>2,Times@@primeMS[#]==Total[primeMS[#]]-1]&]

Formula

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

Extensions

More terms from Jinyuan Wang, Jun 27 2020