A334965 Numbers with strictly increasing prime multiplicities.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169

Offset: 1

Author

Gus Wiseman, May 18 2020

Keywords

• nonn

Comments

First differs from A329131 in lacking 150.

Also numbers whose unsorted prime signature is strictly increasing.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}            25: {3,3}           64: {1,1,1,1,1,1}
    2: {1}           27: {2,2,2}         67: {19}
    3: {2}           29: {10}            71: {20}
    4: {1,1}         31: {11}            73: {21}
    5: {3}           32: {1,1,1,1,1}     75: {2,3,3}
    7: {4}           37: {12}            79: {22}
    8: {1,1,1}       41: {13}            81: {2,2,2,2}
    9: {2,2}         43: {14}            83: {23}
   11: {5}           47: {15}            89: {24}
   13: {6}           49: {4,4}           97: {25}
   16: {1,1,1,1}     50: {1,3,3}         98: {1,4,4}
   17: {7}           53: {16}           101: {26}
   18: {1,2,2}       54: {1,2,2,2}      103: {27}
   19: {8}           59: {17}           107: {28}
   23: {9}           61: {18}           108: {1,1,2,2,2}
		

Crossrefs

These are the Heinz numbers of the partitions counted by A100471.

Partitions with strictly decreasing run-lengths are A100881.

Partitions with weakly decreasing run-lengths are A100882.

Partitions with weakly increasing run-lengths are A100883.

The weakly decreasing version is A242031.

The weakly increasing version is A304678.

The strictly decreasing version is A304686.

Compositions with strictly increasing or decreasing run-lengths are A333191.

Cf. A000837, A032020, A098859, A100883, A329744, A332835, A333147, A333190.

Programs

Mathematica

Select[Range[100],Less@@Last/@FactorInteger[#]&]