A356958 - OEIS

%I #9 Dec 28 2022 09:04:54

%S 1,2,1,1,1,3,1,2,4,2,1,1,1,2,2,1,3,3,5,1,1,2,1,6,1,1,1,4,2,3,1,1,1,1,

%T 4,7,2,1,2,2,8,5,1,1,3,2,4,1,5,1,2,9,1,1,1,2,1,3,3,6,1,6,2,2,2,3,1,1,

%U 4,7,10,1,2,3,11,1,3,1,1,1,1,1,4,2,5

%N Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (b-a+1, ..., y-a+1, z-a+1).

%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.

%e Triangle begins:

%e    1:   .

%e    2:   .

%e    3:   .

%e    4:   1

%e    5:   .

%e    6:   2

%e    7:   .

%e    8:  1 1

%e    9:   1

%e   10:   3

%e   11:   .

%e   12:  1 2

%e   13:   .

%e   14:   4

%e   15:   2

%e   16: 1 1 1

%e For example, the prime indices of 315 are (2,2,3,4), so row 315 is (2,3,4) - 2 + 1 = (1,2,3).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[If[n==1,{},1-First[primeMS[n]]+Rest[primeMS[n]]],{n,100}]

%Y Row lengths are A001222(n) - 1.

%Y Indices of empty rows are A008578.

%Y Even bisection is A112798.

%Y Heinz numbers of rows are A246277.

%Y An opposite version is A358172, Heinz numbers A358195.

%Y Row sums are A359358(n) + A001222(n) - 1.

%Y A112798 list prime indices, sum A056239.

%Y A243055 subtracts the least prime index from the greatest.

%Y Cf. A055396, A124010, A241916, A253565, A325351, A325352, A326844, A355534, A355536, A358137.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Dec 27 2022