A335376 - OEIS

%I #6 Jun 05 2020 09:57:18

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,

%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,

%U 52,53,55,56,57,58,59,61,62,63,64,65,66,67,68,69,70,71

%N Heinz numbers of totally co-strong integer partitions.

%C First differs from A242031 and A317257 in lacking 60.

%C 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.

%C 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.

%e The sequence of terms together with their prime indices begins:

%e     1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}

%e     2: {1}         17: {7}           33: {2,5}

%e     3: {2}         19: {8}           34: {1,7}

%e     4: {1,1}       20: {1,1,3}       35: {3,4}

%e     5: {3}         21: {2,4}         36: {1,1,2,2}

%e     6: {1,2}       22: {1,5}         37: {12}

%e     7: {4}         23: {9}           38: {1,8}

%e     8: {1,1,1}     24: {1,1,1,2}     39: {2,6}

%e     9: {2,2}       25: {3,3}         40: {1,1,1,3}

%e    10: {1,3}       26: {1,6}         41: {13}

%e    11: {5}         27: {2,2,2}       42: {1,2,4}

%e    12: {1,1,2}     28: {1,1,4}       43: {14}

%e    13: {6}         29: {10}          44: {1,1,5}

%e    14: {1,4}       30: {1,2,3}       45: {2,2,3}

%e    15: {2,3}       31: {11}          46: {1,9}

%e 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.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];

%t Select[Range[100],totcostrQ[Reverse[primeMS[#]]]&]

%Y Partitions with weakly increasing run-lengths are A100883.

%Y Totally strong partitions are counted by A316496.

%Y The strong version is A316529.

%Y The version for reversed partitions is (also) A316529.

%Y These partitions are counted by A332275.

%Y The widely normal version is A332293.

%Y The complement is A335377.

%Y Cf. A100882, A133808, A181819, A182850, A242031, A305732, A317256, A317258, A329747, A332291.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 04 2020