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A333219 Heinz number of the n-th composition in standard order.

%I A333219 #8 Mar 17 2020 21:18:10
%S A333219 1,2,3,4,5,6,6,8,7,10,9,12,10,12,12,16,11,14,15,20,15,18,18,24,14,20,
%T A333219 18,24,20,24,24,32,13,22,21,28,25,30,30,40,21,30,27,36,30,36,36,48,22,
%U A333219 28,30,40,30,36,36,48,28,40,36,48,40,48,48,64,17,26,33,44
%N A333219 Heinz number of the n-th composition in standard order.
%C A333219 Includes all positive integers.
%C A333219 The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.
%C A333219 The Heinz number of a composition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A333219 A056239(a(n)) = A070939(n).
%e A333219 The sequence of terms together with their prime indices begins:
%e A333219    1: {}           15: {2,3}          25: {3,3}
%e A333219    2: {1}          20: {1,1,3}        30: {1,2,3}
%e A333219    3: {2}          15: {2,3}          30: {1,2,3}
%e A333219    4: {1,1}        18: {1,2,2}        40: {1,1,1,3}
%e A333219    5: {3}          18: {1,2,2}        21: {2,4}
%e A333219    6: {1,2}        24: {1,1,1,2}      30: {1,2,3}
%e A333219    6: {1,2}        14: {1,4}          27: {2,2,2}
%e A333219    8: {1,1,1}      20: {1,1,3}        36: {1,1,2,2}
%e A333219    7: {4}          18: {1,2,2}        30: {1,2,3}
%e A333219   10: {1,3}        24: {1,1,1,2}      36: {1,1,2,2}
%e A333219    9: {2,2}        20: {1,1,3}        36: {1,1,2,2}
%e A333219   12: {1,1,2}      24: {1,1,1,2}      48: {1,1,1,1,2}
%e A333219   10: {1,3}        24: {1,1,1,2}      22: {1,5}
%e A333219   12: {1,1,2}      32: {1,1,1,1,1}    28: {1,1,4}
%e A333219   12: {1,1,2}      13: {6}            30: {1,2,3}
%e A333219   16: {1,1,1,1}    22: {1,5}          40: {1,1,1,3}
%e A333219   11: {5}          21: {2,4}          30: {1,2,3}
%e A333219   14: {1,4}        28: {1,1,4}        36: {1,1,2,2}
%t A333219 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333219 Table[Times@@Prime/@stc[n],{n,0,100}]
%Y A333219 The length of the k-th composition in standard order is A000120(k).
%Y A333219 The sum of the k-th composition in standard order is A070939(k).
%Y A333219 The maximum of the k-th composition in standard order is A070939(k).
%Y A333219 A partial inverse is A333220. See also A233249.
%Y A333219 Cf. A048793, A056239, A066099, A112798, A114994, A124767, A213925, A225620, A228351, A233564, A272919, A333218.
%K A333219 nonn
%O A333219 1,2
%A A333219 _Gus Wiseman_, Mar 16 2020