A332642 Numbers whose negated unsorted prime signature is not unimodal.

90, 126, 198, 234, 270, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 594, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1188, 1206, 1242, 1274, 1275, 1278, 1314, 1350, 1386

Offset: 1

Author

Gus Wiseman, Feb 28 2020

Keywords

• nonn

Comments

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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}
   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}
   594: {1,2,2,2,5}
   630: {1,2,2,3,4}
   650: {1,3,3,6}
   666: {1,2,2,12}
   702: {1,2,2,2,6}
For example, 630 has negated unsorted prime signature (-1,-2,-1,-1), which is not unimodal, so 630 is in the sequence.
		

Links

• Eric Weisstein's World of Mathematics, Unimodal Sequence

Crossrefs

These are the Heinz numbers of the partitions counted by A332639.

The case that is not unimodal either is A332643.

The version for compositions is A332669.

The complement is A332282.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Unsorted prime signature is A124010.

Non-unimodal normal sequences are A328509.

The number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n is A332742(n).

Partitions whose negated 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.

Cf. A007052, A056239, A112798, A181821, A242031, A329747, A332280, A332281, A332578, A332671, A332831.

Programs

Mathematica

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[2000],!unimodQ[-Last/@FactorInteger[#]]&]