%I #64 Sep 27 2023 15:02:58

%S 1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,1,1,1,

%T 1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,

%U 1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,1,1,0,1,1,1,0,0

%N Leading terms of the rows of the array in A362450; or, Gilbreath transform of tau (A000005).

%C Conjecture: All terms are either 0 or 1. Verified to a(10^7).

%C Inspired by Gilbreath's conjecture, A036262.

%C Using the terminology of A362451, this is the Gilbreath transform of tau (A000005). - _N. J. A. Sloane_, May 05 2023

%H N. J. A. Sloane, <a href="/A361897/b361897.txt">Table of n, a(n) for n = 1..10000</a>

%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.)

%H <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>

%e Table begins (conjecture is leading terms are 0 or 1):

%e 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 ...

%e 1 0 1 1 2 2 2 1 1 2 4 4 2 0 1 3 4 4 4 2 0 2 6 5 1 0 2 4 6 6 4 2 0 0 5 7 2 0 ...

%e 1 1 0 1 0 0 1 0 1 2 0 2 2 1 2 1 0 0 2 2 2 4 1 4 1 2 2 2 0 2 2 2 0 5 2 5 2 4 ...

%e 0 1 1 1 0 1 1 1 1 2 2 0 1 1 1 1 0 2 0 0 2 3 3 3 1 0 0 2 2 0 0 2 5 3 3 3 2 ...

%e 1 0 0 1 1 0 0 0 1 0 2 1 0 0 0 1 2 2 0 2 1 0 0 2 1 0 2 0 2 0 2 3 2 0 0 1 0 ...

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

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

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

%e 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 ...

%e 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 ...

%e 0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 ...

%e 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 ...

%e etc.

%e ...

%e The first two rows are A000005, abs(A051950). The full table, read by antidiagonals, is A362450.

%p N:= 200: # for a(1) to a(N)

%p L:= [seq(numtheory:-tau(n),n=1..N)]:

%p for i from 1 to 105 do

%p R[i]:= L[1];

%p L:= map(abs,L[2..-1]-L[1..-2])

%p od:

%p seq(R[i],i=1..M); # _Robert Israel_, May 07 2023

%t a[n_] := NestWhile[ Abs@ Differences@ # &, Table[ DivisorSigma[0, m], {m, n}], Length[##] > 1 &][[1]]; Array[a, 105]

%t (* or *)

%t mx = 105; lst = {}; k = 0; d = Array[ DivisorSigma[0, #] &, mx]; While[k < mx, AppendTo[lst, d[[1]]]; d = Abs@ Differences@ d; k++]; lst

%t (* or *)

%t A361897[nmax_]:=Module[{d=DivisorSigma[0,Range[nmax]]},Join[{1},Table[First[d=Abs[Differences[d]]],nmax-1]]];A361897[200] (* _Paolo Xausa_, May 07 2023 *)

%o (PARI) lista(nn) = my(v=apply(numdiv, [1..nn]), list = List(), nb=nn); listput(list, v[1]); for (n=2, nn, nb--; my(w = vector(nb, k, abs(v[k+1]-v[k]))); listput(list, w[1]); v = w;); Vec(list);

%o lista(200) \\ _Michel Marcus_, Mar 29 2023

%Y Cf. A000005, A036262, A051950, A362450, A362451, A362452, A362453 (0's), A362454 (1's).

%Y See also A001659 (if don't use absolute values).

%K easy,nonn

%O 1,1

%A Wayman Eduardo Luy and _Robert G. Wilson v_, Mar 28 2023

%E Edited by _N. J. A. Sloane_, Apr 30 2023