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%I #9 Mar 17 2020 21:18:16
%S 0,1,2,3,4,6,8,7,10,12,16,14,32,24,20,15,64,26,128,28,40,48,256,30,36,
%T 96,42,56,512,52,1024,31,80,192,72,58,2048,384,160,60,4096,104,8192,
%U 112,84,768,16384,62,136,100,320,224,32768,106,144,120,640,1536
%N The number k such that the k-th composition in standard order consists of the prime indices of n in weakly increasing order.
%C 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 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%F A000120(a(n)) = A056239(n).
%e The sequence of terms together with the corresponding compositions begins:
%e 0: () 128: (8) 2048: (12)
%e 1: (1) 28: (1,1,3) 384: (1,8)
%e 2: (2) 40: (2,4) 160: (2,6)
%e 3: (1,1) 48: (1,5) 60: (1,1,1,3)
%e 4: (3) 256: (9) 4096: (13)
%e 6: (1,2) 30: (1,1,1,2) 104: (1,2,4)
%e 8: (4) 36: (3,3) 8192: (14)
%e 7: (1,1,1) 96: (1,6) 112: (1,1,5)
%e 10: (2,2) 42: (2,2,2) 84: (2,2,3)
%e 12: (1,3) 56: (1,1,4) 768: (1,9)
%e 16: (5) 512: (10) 16384: (15)
%e 14: (1,1,2) 52: (1,2,3) 62: (1,1,1,1,2)
%e 32: (6) 1024: (11) 136: (4,4)
%e 24: (1,4) 31: (1,1,1,1,1) 100: (1,3,3)
%e 20: (2,3) 80: (2,5) 320: (2,7)
%e 15: (1,1,1,1) 192: (1,7) 224: (1,1,6)
%e 64: (7) 72: (3,4) 32768: (16)
%e 26: (1,2,2) 58: (1,1,2,2) 106: (1,2,2,2)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t Table[fbi[Accumulate[Reverse[primeMS[n]]]],{n,100}]
%Y The version with prime indices taken in weakly decreasing order is A233249.
%Y A partial inverse is A333219.
%Y Cf. A000120, A029931, A048793, A056239, A066099, A070939, A112798, A114994, A225620, A228351, A333221.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 17 2020
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