OFFSET
1,2
COMMENTS
A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong 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.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 16: {1,1,1,1} 32: {1,1,1,1,1}
2: {1} 17: {7} 33: {2,5}
3: {2} 19: {8} 34: {1,7}
4: {1,1} 20: {1,1,3} 35: {3,4}
5: {3} 21: {2,4} 36: {1,1,2,2}
6: {1,2} 22: {1,5} 37: {12}
7: {4} 23: {9} 38: {1,8}
8: {1,1,1} 24: {1,1,1,2} 39: {2,6}
9: {2,2} 25: {3,3} 40: {1,1,1,3}
10: {1,3} 26: {1,6} 41: {13}
11: {5} 27: {2,2,2} 42: {1,2,4}
12: {1,1,2} 28: {1,1,4} 43: {14}
13: {6} 29: {10} 44: {1,1,5}
14: {1,4} 30: {1,2,3} 45: {2,2,3}
15: {2,3} 31: {11} 46: {1,9}
For example, 180 is the Heinz number of (3,2,2,1,1) which has run-lengths: (1,2,2) -> (1,2) -> (1,1) -> (2) -> (1). All of these are weakly increasing, so 180 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totcostrQ[q_]:=Or[Length[q]<=1, And[OrderedQ[Length/@Split[q]], totcostrQ[Length/@Split[q]]]];
Select[Range[100], totcostrQ[Reverse[primeMS[#]]]&]
CROSSREFS
Partitions with weakly increasing run-lengths are A100883.
Totally strong partitions are counted by A316496.
The strong version is A316529.
The version for reversed partitions is (also) A316529.
These partitions are counted by A332275.
The widely normal version is A332293.
The complement is A335377.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 04 2020
STATUS
approved