OFFSET
1,1
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
LINKS
EXAMPLE
The sequence of terms together with their prime indices begins:
90: {1,2,2,3}
126: {1,2,2,4}
180: {1,1,2,2,3}
198: {1,2,2,5}
234: {1,2,2,6}
252: {1,1,2,2,4}
270: {1,2,2,2,3}
306: {1,2,2,7}
342: {1,2,2,8}
350: {1,3,3,4}
360: {1,1,1,2,2,3}
378: {1,2,2,2,4}
396: {1,1,2,2,5}
414: {1,2,2,9}
450: {1,2,2,3,3}
468: {1,1,2,2,6}
504: {1,1,1,2,2,4}
522: {1,2,2,10}
525: {2,3,3,4}
540: {1,1,2,2,2,3}
For example, 350 is the Heinz number of (4,3,3,1), with negated first differences (1,0,2), which is not unimodal, so 350 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Select[Range[1000], !unimodQ[Differences[primeMS[#]]]&]
CROSSREFS
The complement is too full.
The enumeration of these partitions by sum is A332284.
The version where the last part is taken to be 0 is A332832.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Heinz numbers of partitions with weakly increasing differences are A325360.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 26 2020
STATUS
approved