A362463 Array of numbers read by upward antidiagonals: leading row lists the primes as they were in the 19th century (A008578); the following rows give absolute values of differences of previous row.

1, 1, 2, 0, 1, 3, 1, 1, 2, 5, 0, 1, 0, 2, 7, 1, 1, 2, 2, 4, 11, 0, 1, 2, 0, 2, 2, 13, 1, 1, 2, 0, 0, 2, 4, 17, 0, 1, 2, 0, 0, 0, 2, 2, 19, 1, 1, 2, 0, 0, 0, 0, 2, 4, 23, 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29, 1, 1, 0, 2, 2, 2, 2, 2, 2, 4, 2, 31, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 4, 6, 37, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2, 4, 41

Offset: 1

Author

N. J. A. Sloane, May 08 2023

Keywords

• nonn
• tabl

Comments

The Gilbreath transform (cf. A362451) of A008578.

Analogous to A036262. The Gilbreath conjecture is that the initial terms of the rows are 1,(1,0)* = A135528.

Examples

			The array begins:
  1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67
  1 1 2 2 4  2  4  2  4  6  2  6  4  2  4  6  6  2  6  4
  0 1 0 2 2  2  2  2  2  4  4  2  2  2  2  0  4  4  2  2
  1 1 2 0 0  0  0  0  2  0  2  0  0  0  2  4  0  2  0  2
  0 1 2 0 0  0  0  2  2  2  2  0  0  2  2  4  2  2  2  0
  1 1 2 0 0  0  2  0  0  0  2  0  2  0  2  2  0  0  2  2
  0 1 2 0 0  2  2  0  0  2  2  2  2  2  0  2  0  2  0  0
  1 1 2 0 2  0  2  0  2  0  0  0  0  2  2  2  2  2  0  0
  0 1 2 2 2  2  2  2  2  0  0  0  2  0  0  0  0  2  0  0
  1 1 0 0 0  0  0  0  2  0  0  2  2  0  0  0  2  2  0  0
The first few antidiagonals are:
  1,
  1, 2,
  0, 1, 3,
  1, 1, 2, 5,
  0, 1, 0, 2, 7,
  1, 1, 2, 2, 4, 11,
  0, 1, 2, 0, 2, 2, 13,
  1, 1, 2, 0, 0, 2, 4, 17,
  0, 1, 2, 0, 0, 0, 2, 2, 19,
  1, 1, 2, 0, 0, 0, 0, 2, 4, 23,
  0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29,
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
• N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Crossrefs

Cf. A008578, A036262, A135528, A362451.

Programs

Mathematica

A362463[dmax_]:=With[{d=Reverse[NestList[Abs[Differences[#]]&,Join[{1},Prime[Range[dmax-1]]],dmax-1]]},Array[Diagonal[d,#]&,dmax,1-dmax]];A362463[20] (* Generates 20 antidiagonals *) (* Paolo Xausa, May 08 2023 *)