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A362451 Gilbreath transform of {sigma(i), i >= 1} (cf. A000203).

%I A362451 #83 Mar 27 2025 02:23:26
%S A362451 1,2,1,1,1,2,1,0,0,1,0,4,0,3,0,2,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,0,
%T A362451 1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,1,1,
%U A362451 1,0,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0,1,1,1,0,1,1,1,68,0,14,0,7,0,2,0,21,1,8,1,9,1,0,1,18,0,7,0,2,0,1,0,13,1,1,1,2,1,1
%N A362451 Gilbreath transform of {sigma(i), i >= 1} (cf. A000203).
%C A362451 Given a sequence {u(i), i >= o} with offset o, its absolute difference sequence is the sequence {v(i) = |u(i+1)-u(i)|, i >= o}.
%C A362451 The Gilbreath transform of a sequence s = {s(i), i >= o} is constructed as follows.
%C A362451 Form an array A in which the initial row is s and each subsequence row is the absolute difference sequence of the previous row. The sequence of leading terms of the rows of A is the Gilbreath transform of s.
%C A362451 If "absolute difference sequence" is changed to the familiar "first differences", instead of the Gilbreath transform we get the usual inverse binomial transform.
%C A362451 It appears that the terms are mostly 0's and 1's, with occasional eruptions of "geysers". See A362456, A362457.
%H A362451 N. J. A. Sloane, <a href="/A362451/b362451.txt">Table of n, a(n) for n = 1..60000</a>
%H A362451 N. J. A. Sloane, <a href="/A362451/a362451_2.txt">Maple code for Gilbreath transform and related arrays</a>
%H A362451 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A362451 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 A362451 Paolo Xausa, <a href="/A362451/a362451.txt">Table of n, a(n) for n = 1..1000000</a>
%H A362451 Paolo Xausa, <a href="/A362451/a362451_3.png">Logarithmic scatterplot for n = 1..1100000</a>
%H A362451 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%e A362451 We give two examples. (1) For the Gilbreath transform of the sequence of primes (cf. A000040), the array A is given in A036262. The Gilbreath transform begins {2, 1, 1, 1, 1, ...}, and the famous Gilbreath conjecture is that every term after the initial 2 is equal to 1.
%e A362451 (2) For the Gilbreath transform of {tau(i), i >= 1} (cf. A000005), the array A is given in A362450, and the Gilbreath transform is given in A361897. The authors of the latter sequence conjecture that its terms are just 0's and 1's.
%e A362451 See A362452 for a further example.
%p A362451 # To get M terms of the Gilbreath transform of s, assuming offset is 1:
%p A362451 GT := proc(s,M) local G,u,i;
%p A362451 u := [seq(s[i],i=1..M)];
%p A362451 G:=[s[1]];
%p A362451 for i from 1 to M-1 do
%p A362451 u:=[seq(abs(u[i+1]-u[i]),i=1..nops(u)-1)];
%p A362451 G:=[op(G),u[1]]; od:
%p A362451 G;
%p A362451 end;
%p A362451 # For the present sequence:
%p A362451 GT(numtheory[sigma],150);
%p A362451 # See link for a more comprehensive Maple program
%t A362451 A362451[nmax_]:=Module[{d=DivisorSigma[1,Range[nmax]]},Join[{1},Table[First[d=Abs[Differences[d]]],nmax-1]]];A362451[200] (* _Paolo Xausa_, May 07 2023 *)
%o A362451 (PARI)
%o A362451 lista(nn) = my(v=apply(sigma, [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 A362451 lista(200) \\ (based on PARI program in A361897)
%Y A362451 Cf. A000005, A000040, A000203, A036262, A361897, A362450, A362452, A362456, A362457, A362464.
%K A362451 nonn
%O A362451 1,2
%A A362451 _N. J. A. Sloane_, May 03 2023
%E A362451 More than the usual number of terms are displayed in order to go out beyond the long initial 0,1 subsequence.