A375128 - OEIS

%I #7 Aug 05 2024 08:45:58

%S 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 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 1,1,2,2,1,15,1,1,1,1,4,4,1,3,2,1,1,16

%N 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 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 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 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e 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 Triangle begins:

%e    1:

%e    2:  1

%e    3:  2

%e    4:  1  1

%e    5:  3

%e    6:  1

%e    7:  4

%e    8:  1  1  1

%e    9:  2  2

%e   10:  1

%e   11:  5

%e   12:  1  1

%e   13:  6

%e   14:  1

%e   15:  2

%e   16:  1  1  1  1

%t Table[If[n==1,{},First/@Split[Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]],Less]],{n,100}]

%Y Row-minima are A055396.

%Y Row-sums are A374706.

%Y Row-lengths are A375136.

%Y For leaders of constant runs we have A304038, row-sums A066328.

%Y For compositions we have A374683, row-sums of A374684 (length A124768).

%Y A112798 lists prime indices:

%Y - length A001222, distinct A001221

%Y - leader A055396

%Y - sum A056239

%Y - reverse A296150

%Y Cf. A034296, A141199, A218482, A279790, A320324, A333213, A358836, A374634, A374700, A374758, A375133.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Aug 04 2024