login
Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.
2

%I #7 Nov 11 2021 20:05:41

%S 1,2,1,4,2,1,3,6,2,1,4,3,5,8,2,1,6,4,7,3,5,11,3,2,8,5,10,4,6,1,7,9,15,

%T 3,2,11,6,13,5,7,1,9,12,4,8,10,14,18,3,2,14,7,16,6,8,1,10,15,4,9,12,

%U 17,5,11,13,24,4,3,19,10,21,9,11,2,14,20,6,12,16

%N Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.

%C We find k at S(m,k) where S is A349191 read as an irregular triangle. Alternatively, we find prime(k) at U(m,k) where U is A348907 read as an irregular triangle.

%H Michael De Vlieger, <a href="/A349192/b349192.txt">Table of n, a(n) for n = 1..10237</a> (rows 1 <= n <= 35, flattened)

%H Michael De Vlieger, <a href="/A349192/a349192.png">Log-log scatterplot of T(m,k)</a> 1 <= m <= 36.

%F Row lengths are in A338237.

%e First rows of T(m,k):

%e m\k 1 2 3 4 5 6 7 8 9 10 11

%e -----------------------------------------------

%e 1: 1

%e 2: 2 1

%e 3: 4 2 1 3

%e 4: 6 2 1 4 3 5

%e 5: 8 2 1 6 4 7 3 5

%e 6: 11 3 2 8 5 10 4 6 1 7 9

%e ... (End)

%t c = 0; Flatten@ Map[Table[If[k == 1, Length[#] + 1, FirstPosition[#, k - 1][[1]]], {k, If[IntegerQ@ #, # + 1, 1] &@ Max[#]}] &, {{}}~Join~Most@ SplitBy[Reap[Do[Set[a[i], If[PrimeQ[i], i; c++, a[i - c]]]; Sow[a[i]], {i, 2, 100}]][[-1, -1]], # == 0 &][[2 ;; -1 ;; 2]]]

%Y Cf. A000027, A000040, A000720, A338237, A348907, A349191.

%K nonn,tabf

%O 1,2

%A _Michael De Vlieger_, Nov 09 2021