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A362452
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Gilbreath transform of {sigma(i)-i, i >= 1} (see sum of aliquot parts, A001065).
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5
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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, 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, 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
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OFFSET
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1,120
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COMMENTS
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See A362451 for further information.
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.
[It would be nice to have plots of larger numbers of initial terms.]
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LINKS
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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.)
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MAPLE
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# To get M terms of the Gilbreath transform of s:
GT := proc(s, M) local G, u, i;
u := [seq(s(i), i=1..M)];
G:=[s(1)];
for i from 1 to M-1 do
u:=[seq(abs(u[i+1]-u[i]), i=1..nops(u)-1)];
G:=[op(G), u[1]]; od:
G;
end;
# For the present sequence:
aliq := proc(n) numtheory[sigma](n) - n; end;
GT(aliq, 150);
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MATHEMATICA
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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 *)
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PROG
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(PARI)
f(n) = sigma(n) - n
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);
lista(200)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More than the usual number of terms are displayed in order to go out beyond the long initial 0,1 subsequence.
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STATUS
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approved
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