A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

1, 2, 4, 6, 8, 16, 18, 20, 32, 54, 56, 64, 100, 128, 162, 176, 256, 392, 416, 486, 500, 512, 1024, 1088, 1458, 1936, 2048, 2432, 2500, 2744, 4096, 4374, 5408, 5888, 8192, 12500, 13122, 14848, 16384, 18496, 19208, 21296, 31744, 32768, 39366, 46208, 62500, 65536

Offset: 1

Author

Gus Wiseman, May 20 2019

Keywords

• nonn

Comments

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

The enumeration of these partitions by sum is given by A325765.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   256: {1,1,1,1,1,1,1,1}
   392: {1,1,1,4,4}
   416: {1,1,1,1,1,6}
   486: {1,2,2,2,2,2}
   500: {1,1,3,3,3}
   512: {1,1,1,1,1,1,1,1,1}
		

Crossrefs

Cf. A002033, A056239, A103295, A103300, A112798, A143823, A169942, A325676, A325685, A325763, A325765, A325769, A325770.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{_,s__,_}:>{s}]]&&Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]]&]