A332282 Numbers whose unsorted prime signature is not unimodal.

300, 588, 600, 980, 1176, 1200, 1452, 1500, 1960, 2028, 2100, 2205, 2352, 2400, 2420, 2904, 2940, 3000, 3300, 3380, 3388, 3468, 3900, 3920, 4056, 4116, 4200, 4332, 4410, 4704, 4732, 4800, 4840, 5100, 5445, 5700, 5780, 5808, 5880, 6000, 6348, 6468, 6600, 6615

Offset

1,1

Comments

The unsorted prime signature of a positive integer (row n of A124010) is the sequence of exponents in its prime factorization.

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

Also Heinz numbers of integer partitions with non-unimodal run-lengths. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

From Michael De Vlieger, Jan 28 2026: (Start)

Subsequence of both A000977 and A059404.

Define an m-full number k to be such that exponents m >= k for all prime power factors p^m | k, but p^(m+1) does not divide k. The smallest m-full number k in this sequence is 2^(m+1) * 3^m * 5^(m+1). Thus the smallest 2-full (powerful) term is a(66) = 9000 = 2^3 * 3^2 * 5^3, the smallest 3-full (cubefull) term is a(2445) = 270000, etc. (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Example

The sequence of terms together with their prime indices begins:
   300: {1,1,2,3,3}
   588: {1,1,2,4,4}
   600: {1,1,1,2,3,3}
   980: {1,1,3,4,4}
  1176: {1,1,1,2,4,4}
  1200: {1,1,1,1,2,3,3}
  1452: {1,1,2,5,5}
  1500: {1,1,2,3,3,3}
  1960: {1,1,1,3,4,4}
  2028: {1,1,2,6,6}
  2100: {1,1,2,3,3,4}
  2205: {2,2,3,4,4}
  2352: {1,1,1,1,2,4,4}
  2400: {1,1,1,1,1,2,3,3}
  2420: {1,1,3,5,5}
  2904: {1,1,1,2,5,5}
  2940: {1,1,2,3,4,4}
  3000: {1,1,1,2,3,3,3}
  3300: {1,1,2,3,3,5}
  3380: {1,1,3,6,6}

Mathematica

unimodQ[q_]:=Or[Length[q]==1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Select[Range[1000], !unimodQ[Last/@FactorInteger[#]]&]

Crossrefs

The opposite version is A332642.

These are the Heinz numbers of the partitions counted by A332281.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Cf. A000977, A001523, A007052, A056239, A059404, A112798, A124010, A227038, A332280, A332284, A332286, A332639, A332643, A332671.

Sequence in context: A116346 A102509 A250008 * A190879 A375384 A154061

Adjacent sequences: A332279 A332280 A332281 * A332283 A332284 A332285

Keyword

nonn

Author

Gus Wiseman, Feb 19 2020

Status

approved