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A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

%I A358172 #9 Dec 28 2022 09:04:58
%S A358172 1,2,1,1,1,3,2,2,4,2,1,1,1,2,1,3,3,3,5,2,2,2,1,6,1,1,4,4,3,2,1,1,1,1,
%T A358172 4,7,2,2,2,1,8,5,3,3,3,4,3,5,5,2,2,9,2,2,2,2,1,3,1,6,6,6,2,1,1,3,4,4,
%U A358172 4,7,10,3,3,2,11,3,3,1,1,1,1,1,4,5,4
%N A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).
%C A358172 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 A358172 Triangle begins:
%e A358172    1:   .
%e A358172    2:   .
%e A358172    3:   .
%e A358172    4:   1
%e A358172    5:   .
%e A358172    6:   2
%e A358172    7:   .
%e A358172    8:  1 1
%e A358172    9:   1
%e A358172   10:   3
%e A358172   11:   .
%e A358172   12:  2 2
%e A358172   13:   .
%e A358172   14:   4
%e A358172   15:   2
%e A358172   16: 1 1 1
%e A358172   17:   .
%e A358172   18:  2 1
%e A358172   19:   .
%e A358172   20:  3 3
%e A358172 For example, the prime indices of 900 are (1,1,2,2,3,3), so row 900 is 3 - (1,1,2,2,3) + 1 = (3,3,2,2,1).
%t A358172 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358172 Table[If[n==1,{},1+Last[primeMS[n]]-Most[primeMS[n]]],{n,100}]
%Y A358172 Row lengths are A001222(n) - 1.
%Y A358172 Indices of empty rows are A008578.
%Y A358172 Even-indexed rows have sums A243503.
%Y A358172 Row sums are A326844(n) + A001222(n) - 1.
%Y A358172 An opposite version is A356958, Heinz numbers A246277.
%Y A358172 Heinz numbers of the rows are A358195, even bisection A241916.
%Y A358172 A112798 list prime indices, sum A056239.
%Y A358172 A243055 subtracts the least prime index from the greatest.
%Y A358172 Cf. A055396, A124010, A253565, A325351, A325352, A355534, A355536, A358137.
%K A358172 nonn,tabf
%O A358172 1,2
%A A358172 _Gus Wiseman_, Dec 20 2022