A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172

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 whose product of prime indices (A003963) is less than their sum of prime indices (A056239).

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

Examples

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  46: {1,9}
  48: {1,1,1,1,2}
		

Crossrefs

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]
				

Formula

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

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