A362463 - OEIS

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.

%I #13 Sep 27 2023 15:02:30

%S 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,

%T 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,

%U 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

%N 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.

%C The Gilbreath transform (cf. A362451) of A008578.

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

%H Paolo Xausa, <a href="/A362463/b362463.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened)

%H 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: <a href="https://vimeo.com/866583736?share=copy">Video</a>, <a href="http://neilsloane.com/doc/EMSep2023.pdf">Slides</a>, <a href="http://neilsloane.com/doc/EMSep2023.Updates.txt">Updates</a>. (Mentions this sequence.)

%e The array begins:

%e 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67

%e 1 1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4

%e 0 1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 4 2 2

%e 1 1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 0 2 0 2

%e 0 1 2 0 0 0 0 2 2 2 2 0 0 2 2 4 2 2 2 0

%e 1 1 2 0 0 0 2 0 0 0 2 0 2 0 2 2 0 0 2 2

%e 0 1 2 0 0 2 2 0 0 2 2 2 2 2 0 2 0 2 0 0

%e 1 1 2 0 2 0 2 0 2 0 0 0 0 2 2 2 2 2 0 0

%e 0 1 2 2 2 2 2 2 2 0 0 0 2 0 0 0 0 2 0 0

%e 1 1 0 0 0 0 0 0 2 0 0 2 2 0 0 0 2 2 0 0

%e The first few antidiagonals are:

%e 1,

%e 1, 2,

%e 0, 1, 3,

%e 1, 1, 2, 5,

%e 0, 1, 0, 2, 7,

%e 1, 1, 2, 2, 4, 11,

%e 0, 1, 2, 0, 2, 2, 13,

%e 1, 1, 2, 0, 0, 2, 4, 17,

%e 0, 1, 2, 0, 0, 0, 2, 2, 19,

%e 1, 1, 2, 0, 0, 0, 0, 2, 4, 23,

%e 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29,

%t 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 *)

%Y Cf. A008578, A036262, A135528, A362451.

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, May 08 2023