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A349192 Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.

%I A349192 #10 Aug 17 2025 02:05:10
%S A349192 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 A349192 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 A349192 17,5,11,13,24,4,3,19,10,21,9,11,2,14,20,6,12,16
%N A349192 Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.
%C A349192 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 A349192 Michael De Vlieger, <a href="/A349192/b349192.txt">Table of n, a(n) for n = 1..10237</a> (rows 1 <= n <= 35, flattened)
%H A349192 Michael De Vlieger, <a href="/A349192/a349192.png">Log-log scatterplot of T(m,k)</a> 1 <= m <= 36.
%F A349192 Row lengths are in A338237.
%e A349192 First rows of T(m,k):
%e A349192   m\k   1   2   3   4   5   6   7   8   9  10  11
%e A349192   -----------------------------------------------
%e A349192   1:    1
%e A349192   2:    2   1
%e A349192   3:    4   2   1   3
%e A349192   4:    6   2   1   4   3   5
%e A349192   5:    8   2   1   6   4   7   3   5
%e A349192   6:   11   3   2   8   5  10   4   6   1   7   9
%e A349192   ...
%t A349192 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 A349192 Cf. A000027, A000040, A000720, A338237, A348907, A349191.
%K A349192 nonn,tabf
%O A349192 1,2
%A A349192 _Michael De Vlieger_, Nov 09 2021