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A335377
Heinz numbers of non-totally co-strong integer partitions.
1
18, 50, 54, 60, 75, 84, 90, 98, 108, 120, 126, 132, 140, 147, 150, 156, 162, 168, 198, 204, 220, 228, 234, 240, 242, 245, 250, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 324, 336, 338, 340, 342, 348, 350, 363, 364, 372, 375, 378, 380, 408, 414, 420
OFFSET
1,1
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:
18: {1,2,2} 156: {1,1,2,6} 276: {1,1,2,9}
50: {1,3,3} 162: {1,2,2,2,2} 280: {1,1,1,3,4}
54: {1,2,2,2} 168: {1,1,1,2,4} 294: {1,2,4,4}
60: {1,1,2,3} 198: {1,2,2,5} 300: {1,1,2,3,3}
75: {2,3,3} 204: {1,1,2,7} 306: {1,2,2,7}
84: {1,1,2,4} 220: {1,1,3,5} 308: {1,1,4,5}
90: {1,2,2,3} 228: {1,1,2,8} 312: {1,1,1,2,6}
98: {1,4,4} 234: {1,2,2,6} 315: {2,2,3,4}
108: {1,1,2,2,2} 240: {1,1,1,1,2,3} 324: {1,1,2,2,2,2}
120: {1,1,1,2,3} 242: {1,5,5} 336: {1,1,1,1,2,4}
126: {1,2,2,4} 245: {3,4,4} 338: {1,6,6}
132: {1,1,2,5} 250: {1,3,3,3} 340: {1,1,3,7}
140: {1,1,3,4} 260: {1,1,3,6} 342: {1,2,2,8}
147: {2,4,4} 264: {1,1,1,2,5} 348: {1,1,2,10}
150: {1,2,3,3} 270: {1,2,2,2,3} 350: {1,3,3,4}
For example, 60 is the Heinz number of (3,2,1,1), which has run-lengths: (1,1,2) -> (2,1) -> (1,1) -> (2) -> (1). Since (2,1) is not weakly increasing, 60 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 counted by A100883.
Totally strong partitions are counted by A316496.
Heinz numbers of totally strong partitions are A316529.
The version for reversed partitions is A316597.
The strong version is (also) A316597.
The alternating version is A317258.
Totally co-strong partitions are counted by A332275.
The complement is A335376.
Sequence in context: A135189 A178398 A222740 * A317258 A071365 A360252
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 05 2020
STATUS
approved