A332831 Numbers whose unsorted prime signature is neither weakly increasing nor weakly decreasing.

90, 126, 198, 234, 270, 300, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 588, 594, 600, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 980, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1176, 1188, 1200, 1206, 1242

Offset: 1

Author

Gus Wiseman, Mar 02 2020

Keywords

• nonn

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Examples

			The sequence of terms together with their prime indices begins:
   90: {1,2,2,3}
  126: {1,2,2,4}
  198: {1,2,2,5}
  234: {1,2,2,6}
  270: {1,2,2,2,3}
  300: {1,1,2,3,3}
  306: {1,2,2,7}
  342: {1,2,2,8}
  350: {1,3,3,4}
  378: {1,2,2,2,4}
  414: {1,2,2,9}
  522: {1,2,2,10}
  525: {2,3,3,4}
  540: {1,1,2,2,2,3}
  550: {1,3,3,5}
  558: {1,2,2,11}
  588: {1,1,2,4,4}
  594: {1,2,2,2,5}
  600: {1,1,1,2,3,3}
  630: {1,2,2,3,4}
For example, the prime signature of 540 is (2,3,1), so 540 is in the sequence.
		

Crossrefs

The version for run-lengths of partitions is A332641.

The version for run-lengths of compositions is A332833.

The version for compositions is A332834.

Prime signature is A124010.

Unimodal compositions are A001523.

Partitions with weakly increasing run-lengths are A100883.

Partitions with weakly increasing or decreasing run-lengths are A332745.

Compositions with weakly increasing or decreasing run-lengths are A332835.

Compositions with weakly increasing run-lengths are A332836.

Cf. A100882, A115981, A328509, A329398, A332281, A332640, A332643, A332746, A332836, A332870.

Programs

Mathematica

Select[Range[1000],!Or[LessEqual@@Last/@FactorInteger[#],GreaterEqual@@Last/@FactorInteger[#]]&]

Formula

Intersection of A071365 and A112769.