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

1,3

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.

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.)

Example

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,

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 *)

Crossrefs

Cf. A008578, A036262, A135528, A362451.

Sequence in context: A291316 A130504 A044942 * A263401 A369814 A239928

Adjacent sequences: A362460 A362461 A362462 * A362464 A362465 A362466

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 08 2023

Status

approved