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%I A317257 #13 Jun 06 2020 10:47:03
%S A317257 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 A317257 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,
%U A317257 52,53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N A317257 Heinz numbers of alternately co-strong integer partitions.
%C A317257 The first term absent from this sequence but present in A242031 is 180.
%C A317257 A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
%C A317257 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A317257 The sequence of terms together with their prime indices begins:
%e A317257     1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
%e A317257     2: {1}         17: {7}           33: {2,5}
%e A317257     3: {2}         19: {8}           34: {1,7}
%e A317257     4: {1,1}       20: {1,1,3}       35: {3,4}
%e A317257     5: {3}         21: {2,4}         36: {1,1,2,2}
%e A317257     6: {1,2}       22: {1,5}         37: {12}
%e A317257     7: {4}         23: {9}           38: {1,8}
%e A317257     8: {1,1,1}     24: {1,1,1,2}     39: {2,6}
%e A317257     9: {2,2}       25: {3,3}         40: {1,1,1,3}
%e A317257    10: {1,3}       26: {1,6}         41: {13}
%e A317257    11: {5}         27: {2,2,2}       42: {1,2,4}
%e A317257    12: {1,1,2}     28: {1,1,4}       43: {14}
%e A317257    13: {6}         29: {10}          44: {1,1,5}
%e A317257    14: {1,4}       30: {1,2,3}       45: {2,2,3}
%e A317257    15: {2,3}       31: {11}          46: {1,9}
%t A317257 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A317257 totincQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totincQ[Reverse[Length/@Split[q]]]]];
%t A317257 Select[Range[100],totincQ[Reverse[primeMS[#]]]&]
%Y A317257 Cf. A056239, A100883, A181819, A182850, A242031, A296150, A305732, A317246.
%Y A317257 These partitions are counted by A317256.
%Y A317257 The complement is A317258.
%Y A317257 Totally co-strong partitions are counted by A332275.
%Y A317257 Alternately co-strong compositions are counted by A332338.
%Y A317257 Alternately co-strong reversed partitions are counted by A332339.
%Y A317257 The total version is A335376.
%Y A317257 Cf. A182857, A304660, A305563, A316496, A332292, A332340.
%K A317257 nonn
%O A317257 1,2
%A A317257 _Gus Wiseman_, Jul 25 2018
%E A317257 Updated with corrected terminology by _Gus Wiseman_, Jun 04 2020

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