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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, 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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, May 08 2023
STATUS
approved