OFFSET
1,1
COMMENTS
Conjecture: All terms are either 0 or 1. Verified to a(10^7).
Inspired by Gilbreath's conjecture, A036262.
Using the terminology of A362451, this is the Gilbreath transform of tau (A000005). - N. J. A. Sloane, May 05 2023
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..10000
EXAMPLE
Table begins (conjecture is leading terms are 0 or 1):
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
etc.
...
MAPLE
N:= 200: # for a(1) to a(N)
L:= [seq(numtheory:-tau(n), n=1..N)]:
for i from 1 to 105 do
R[i]:= L[1];
L:= map(abs, L[2..-1]-L[1..-2])
od:
seq(R[i], i=1..M); # Robert Israel, May 07 2023
MATHEMATICA
a[n_] := NestWhile[ Abs@ Differences@ # &, Table[ DivisorSigma[0, m], {m, n}], Length[##] > 1 &][[1]]; Array[a, 105]
(* or *)
mx = 105; lst = {}; k = 0; d = Array[ DivisorSigma[0, #] &, mx]; While[k < mx, AppendTo[lst, d[[1]]]; d = Abs@ Differences@ d; k++]; lst
(* or *)
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 *)
PROG
(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);
lista(200) \\ Michel Marcus, Mar 29 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wayman Eduardo Luy and Robert G. Wilson v, Mar 28 2023
EXTENSIONS
Edited by N. J. A. Sloane, Apr 30 2023
STATUS
approved