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A341970 Irregular triangle read by rows of the indices of the entries of 1's in the corresponding rows in A237048.

%I A341970 #11 Mar 11 2021 21:28:21
%S A341970 1,1,1,2,1,1,2,1,3,1,2,1,1,2,3,1,4,1,2,1,3,1,2,1,4,1,2,3,5,1,1,2,1,3,
%T A341970 4,1,2,1,5,1,2,3,6,1,4,1,2,1,3,1,2,5,1,4,1,2,3,6,1,7,1,2,1,3,4,5,1,2,
%U A341970 1,1,2,3,6,1,4,1,2,5,7,1,3,8,1,2,1,4,1,2,3,6,1,5
%N A341970 Irregular triangle read by rows of the indices of the entries of 1's in the corresponding rows in A237048.
%C A341970 The number of entries in the n-th row of the table of this sequence is A001227(n), the number of odd divisors of n.
%C A341970 Let number  n = 2^s * q with s >= 0 and q odd, let row(n) = floor( (sqrt(8*n+1) - 1)/2 ), let D_n = { d : d odd divisor of n and d <= row(n) }, let E_n = { e : e = 2^(s+1) * d,  d in D_n and e <= row(n) } and let F_n be the union of D_n and E_n with its elements listed in increasing order. Then the numbers in F_n are exactly the positions of 1's in row n of A237048 and the numbers in row n of this sequence.
%F A341970 Let 1 <= n, 1 <= d <= A001227(n) and k the index of the d-th 1 in row n of A237048.
%F A341970 a( A060831(n-1) + d ) = T(n, d) = k.
%e A341970 a(8, 9) = { 1, 3 } is row 6 in this sequence with corresponding row 6  { 1,  0,  1 } in A237048.
%e A341970 a(26...29) = { 1, 2, 3, 5 } is row 15 in this sequence with corresponding row 15 { 1, 1, 1, 0, 1 } in A237048.
%e A341970 Table of the first 15 rows:
%e A341970 row    entries
%e A341970 1      1
%e A341970 2      1
%e A341970 3      1  1
%e A341970 4      1
%e A341970 5      1  2
%e A341970 6      1  3
%e A341970 7      1  2
%e A341970 8      1
%e A341970 9      1  2  3
%e A341970 10     1  4
%e A341970 11     1  2
%e A341970 12     1  3
%e A341970 13     1  2
%e A341970 14     1  4
%e A341970 15     1  2  3  5
%t A341970 row[n_] := Floor[(Sqrt[8 n+1]-1)/2]
%t A341970 oddD[n_] := Select[Divisors[n], OddQ[#]&&#<=row[n]&]
%t A341970 twoExp[n_] := Module[{f=FactorInteger[n]}, If[First[First[f]]==2, Last[First[f]], 0]]
%t A341970 dualD[n_] := Select[Map[2^(twoExp[n]+1)#&, oddD[n]], #<=row[n]&]
%t A341970 a341970[n_] := Union[oddD[n], dualD[n]]
%t A341970 Flatten[Map[a341970, Range[40]]] (* first 40 rows of table *)
%Y A341970 Cf. A001227, A060831, A237048, A237593.
%K A341970 nonn,tabf
%O A341970 1,4
%A A341970 _Hartmut F. W. Hoft_, Feb 24 2021