A362452 - OEIS

%I #47 Sep 27 2023 15:04:18

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

%T 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 1,0,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0,1,1,1,0,1,1,1,62,0,12,0,3,0,2,0,25,1

%N Gilbreath transform of {sigma(i)-i, i >= 1} (see sum of aliquot parts, A001065).

%C See A362451 for further information.

%C The first 50000 terms of the present sequence suggest that the terms are usually 0's and 1's, except for occasional "geysers". See A362458, A362459.

%C [It would be nice to have plots of larger numbers of initial terms.]

%H N. J. A. Sloane, <a href="/A362452/b362452.txt">Table of n, a(n) for n = 1..50000</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 Paolo Xausa, <a href="/A362452/a362452.txt">Table of n, a(n) for n = 1..1000000</a>

%H Paolo Xausa, <a href="/A362452/a362452.png">Logarithmic scatterplot for n = 1..1000000</a>

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

%p # To get M terms of the Gilbreath transform of s:

%p GT := proc(s,M) local G,u,i;

%p u := [seq(s(i),i=1..M)];

%p G:=[s(1)];

%p for i from 1 to M-1 do

%p u:=[seq(abs(u[i+1]-u[i]),i=1..nops(u)-1)];

%p G:=[op(G),u[1]]; od:

%p G;

%p end;

%p # For the present sequence:

%p aliq := proc(n) numtheory[sigma](n) - n; end;

%p GT(aliq,150);

%t A362452[nmax_]:=Module[{d=DivisorSigma[1,Range[nmax]]-Range[nmax]},Join[{0},Table[First[d=Abs[Differences[d]]],nmax-1]]];A362452[200] (* _Paolo Xausa_, May 07 2023 *)

%o (PARI)

%o f(n) = sigma(n) - n

%o lista(nn) = my(v=apply(f, [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)

%Y Cf. A000005, A000040, A000203, A001065, A036262, A361897, A362450, A362451, A362458, A362459.

%K nonn

%O 1,120

%A _N. J. A. Sloane_, May 03 2023

%E More than the usual number of terms are displayed in order to go out beyond the long initial 0,1 subsequence.