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A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

%I A375128 #7 Aug 05 2024 08:45:58
%S A375128 1,2,1,1,3,1,4,1,1,1,2,2,1,5,1,1,6,1,2,1,1,1,1,7,1,2,8,1,1,2,1,9,1,1,
%T A375128 1,3,3,1,2,2,2,1,1,10,1,11,1,1,1,1,1,2,1,3,1,1,2,12,1,2,1,1,1,13,1,14,
%U A375128 1,1,2,2,1,15,1,1,1,1,4,4,1,3,2,1,1,16
%N A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.
%C A375128 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.
%C A375128 The minima of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%H A375128 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A375128 The prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), which is row 540.
%e A375128 Triangle begins:
%e A375128    1:
%e A375128    2:  1
%e A375128    3:  2
%e A375128    4:  1  1
%e A375128    5:  3
%e A375128    6:  1
%e A375128    7:  4
%e A375128    8:  1  1  1
%e A375128    9:  2  2
%e A375128   10:  1
%e A375128   11:  5
%e A375128   12:  1  1
%e A375128   13:  6
%e A375128   14:  1
%e A375128   15:  2
%e A375128   16:  1  1  1  1
%t A375128 Table[If[n==1,{},First/@Split[Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]],Less]],{n,100}]
%Y A375128 Row-minima are A055396.
%Y A375128 Row-sums are A374706.
%Y A375128 Row-lengths are A375136.
%Y A375128 For leaders of constant runs we have A304038, row-sums A066328.
%Y A375128 For compositions we have A374683, row-sums of A374684 (length A124768).
%Y A375128 A112798 lists prime indices:
%Y A375128 - length A001222, distinct A001221
%Y A375128 - leader A055396
%Y A375128 - sum A056239
%Y A375128 - reverse A296150
%Y A375128 Cf. A034296, A141199, A218482, A279790, A320324, A333213, A358836, A374634, A374700, A374758, A375133.
%K A375128 nonn,tabf
%O A375128 1,2
%A A375128 _Gus Wiseman_, Aug 04 2024