A325621 Heinz numbers of integer partitions whose reciprocal factorial sum is an integer.

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 128, 144, 162, 256, 288, 324, 375, 512, 576, 648, 729, 750, 1024, 1152, 1296, 1458, 1500, 2048, 2304, 2592, 2916, 3000, 3375, 4096, 4608, 5184, 5832, 6000, 6561, 6750, 8192, 9216, 10368, 11664, 12000, 13122, 13500

Offset: 1

Author

Gus Wiseman, May 13 2019

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
      9: {2,2}
     16: {1,1,1,1}
     18: {1,2,2}
     32: {1,1,1,1,1}
     36: {1,1,2,2}
     64: {1,1,1,1,1,1}
     72: {1,1,1,2,2}
     81: {2,2,2,2}
    128: {1,1,1,1,1,1,1}
    144: {1,1,1,1,2,2}
    162: {1,2,2,2,2}
    256: {1,1,1,1,1,1,1,1}
    288: {1,1,1,1,1,2,2}
    324: {1,1,2,2,2,2}
    375: {2,3,3,3}
    512: {1,1,1,1,1,1,1,1,1}
		

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A058360, A316856, A325619, A325620, A325623.

Programs

Mathematica

Select[Range[1000],IntegerQ[Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]