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Irregular triangle read by rows where row n lists the set of STC-numbers of permutations of the prime indices of n.
24

%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