A325371 Numbers whose prime signature has multiplicities of its parts all distinct and covering an initial interval of positive integers.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 126, 127, 128, 131, 132, 137, 139, 140, 149, 150, 151, 156, 157, 163

Offset: 1

Author

Gus Wiseman, May 02 2019

Keywords

• nonn

Comments

The first term that is not 1 or a prime power is 60.

The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

Numbers whose prime signature has distinct parts that cover an initial interval are given by A325337.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325331.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}
		

Crossrefs

Cf. A055932, A056239, A098859, A112798, A118914, A130091, A317090, A325329, A325330, A325331, A325337, A325369, A325370.

Programs

Mathematica

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&&UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]