|
%I #8 Mar 17 2020 21:18:23
%S 0,1,2,3,4,5,6,8,7,10,9,12,16,11,13,14,32,17,24,18,20,15,64,21,22,26,
%T 128,19,25,28,34,40,33,48,256,23,27,29,30,36,65,96,42,35,49,56,512,37,
%U 38,41,44,50,52,1024,31,66,80,129,192,68,72,43,45,46,53,54,58
%N Irregular triangle read by rows where row n lists the set of STC-numbers of permutations of the prime indices of n.
%C This is a permutation of the nonnegative integers.
%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. We define the composition with STC-number k to be the k-th composition in standard order.
%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.
%e Reading by columns gives:
%e 0 1 2 3 4 5 8 7 10 9 16 11 32 17 18 15 64 21 128 19
%e 6 12 13 24 20 22 25
%e 14 26 28
%e 34 33 256 23 36 65 42 35 512 37 1024 31 66 129 68 43
%e 40 48 27 96 49 38 80 192 72 45
%e 29 56 41 46
%e 30 44 53
%e 50 54
%e 52 58
%e The sequence of terms together with the corresponding compositions begins:
%e 0: () 24: (1,4) 27: (1,2,1,1)
%e 1: (1) 18: (3,2) 29: (1,1,2,1)
%e 2: (2) 20: (2,3) 30: (1,1,1,2)
%e 3: (1,1) 15: (1,1,1,1) 36: (3,3)
%e 4: (3) 64: (7) 65: (6,1)
%e 5: (2,1) 21: (2,2,1) 96: (1,6)
%e 6: (1,2) 22: (2,1,2) 42: (2,2,2)
%e 8: (4) 26: (1,2,2) 35: (4,1,1)
%e 7: (1,1,1) 128: (8) 49: (1,4,1)
%e 10: (2,2) 19: (3,1,1) 56: (1,1,4)
%e 9: (3,1) 25: (1,3,1) 512: (10)
%e 12: (1,3) 28: (1,1,3) 37: (3,2,1)
%e 16: (5) 34: (4,2) 38: (3,1,2)
%e 11: (2,1,1) 40: (2,4) 41: (2,3,1)
%e 13: (1,2,1) 33: (5,1) 44: (2,1,3)
%e 14: (1,1,2) 48: (1,5) 50: (1,3,2)
%e 32: (6) 256: (9) 52: (1,2,3)
%e 17: (4,1) 23: (2,1,1,1) 1024: (11)
%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[Sort[fbi/@Accumulate/@Permutations[primeMS[n]]],{n,30}]
%Y Row lengths are A008480.
%Y Column k = 1 is A233249.
%Y Column k = -1 is A333220.
%Y A related triangle for partitions is A215366.
%Y Cf. A000120, A029931, A048793, A056239, A066099, A070939, A112798, A114994, A225620, A228351, A333218, A333219.
%K nonn,tabf
%O 1,3
%A _Gus Wiseman_, Mar 17 2020
|
