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.

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, 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, 1, 3, 19, 2, 7, 1, 2, 1, 20, 21, 1, 11, 4, 1, 1, 1, 4, 22, 1, 12, 23, 3, 4

Offset: 1

Author

Gus Wiseman, May 23 2025

Keywords

• nonn
• tabf

Comments

All rows are different.

Examples

			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).
The squarefree numbers and corresponding rows begin:
    1: ()        23: (9)        47: (15)
    2: (1)       26: (1,5)      51: (2,5)
    3: (2)       29: (10)       53: (16)
    5: (3)       30: (1,1,1)    55: (3,2)
    6: (1,1)     31: (11)       57: (2,6)
    7: (4)       33: (2,3)      58: (1,9)
   10: (1,2)     34: (1,6)      59: (17)
   11: (5)       35: (3,1)      61: (18)
   13: (6)       37: (12)       62: (1,10)
   14: (1,3)     38: (1,7)      65: (3,3)
   15: (2,1)     39: (2,4)      66: (1,1,3)
   17: (7)       41: (13)       67: (19)
   19: (8)       42: (1,1,2)    69: (2,7)
   21: (2,2)     43: (14)       70: (1,2,1)
   22: (1,4)     46: (1,8)      71: (20)
		

Crossrefs

Row-lengths are A072047, sums A243290.

This is the restriction of A383534 (ranked by A383535) to rows of squarefree index.

A000040 lists the primes, differences A001223.

A048767 is the Look-and-Say transform, union A351294, complement A351295.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.

A325324 counts partitions with distinct 0-appended differences, ranks A325367.

Cf. A001221, A005117, A055396, A061395, A325325, A355536, A358137, A384009.

Programs

Mathematica

sql=Select[Range[100],SquareFreeQ];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[prix[sql[[n]]],0]],{n,Length[sql]}]