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A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.

%I A384008 #5 May 23 2025 10:15:08
%S A384008 1,2,3,1,1,4,1,2,5,6,1,3,2,1,7,8,2,2,1,4,9,1,5,10,1,1,1,11,2,3,1,6,3,
%T A384008 1,12,1,7,2,4,13,1,1,2,14,1,8,15,2,5,16,3,2,2,6,1,9,17,18,1,10,3,3,1,
%U A384008 1,3,19,2,7,1,2,1,20,21,1,11,4,1,1,1,4,22,1,12,23,3,4
%N A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.
%C A384008 All rows are different.
%e A384008 The 28-th squarefree number is 42, with 0-prepended prime indices (0,1,2,4), with differences (1,1,2), so row 28 is (1,1,2).
%e A384008 The squarefree numbers and corresponding rows begin:
%e A384008     1: ()        23: (9)        47: (15)
%e A384008     2: (1)       26: (1,5)      51: (2,5)
%e A384008     3: (2)       29: (10)       53: (16)
%e A384008     5: (3)       30: (1,1,1)    55: (3,2)
%e A384008     6: (1,1)     31: (11)       57: (2,6)
%e A384008     7: (4)       33: (2,3)      58: (1,9)
%e A384008    10: (1,2)     34: (1,6)      59: (17)
%e A384008    11: (5)       35: (3,1)      61: (18)
%e A384008    13: (6)       37: (12)       62: (1,10)
%e A384008    14: (1,3)     38: (1,7)      65: (3,3)
%e A384008    15: (2,1)     39: (2,4)      66: (1,1,3)
%e A384008    17: (7)       41: (13)       67: (19)
%e A384008    19: (8)       42: (1,1,2)    69: (2,7)
%e A384008    21: (2,2)     43: (14)       70: (1,2,1)
%e A384008    22: (1,4)     46: (1,8)      71: (20)
%t A384008 sql=Select[Range[100],SquareFreeQ];
%t A384008 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384008 Table[Differences[Prepend[prix[sql[[n]]],0]],{n,Length[sql]}]
%Y A384008 Row-lengths are A072047, sums A243290.
%Y A384008 This is the restriction of A383534 (ranked by A383535) to rows of squarefree index.
%Y A384008 A000040 lists the primes, differences A001223.
%Y A384008 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A384008 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A384008 A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
%Y A384008 A325324 counts partitions with distinct 0-appended differences, ranks A325367.
%Y A384008 Cf. A001221, A005117, A055396, A061395, A325325, A355536, A358137, A384009.
%K A384008 nonn,tabf
%O A384008 1,2
%A A384008 _Gus Wiseman_, May 23 2025