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A054978
Obtained from sequence of lucky numbers (A000959) by taking iterated absolute value differences of terms and extracting the leading diagonal.
5
1, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 2
OFFSET
0,2
COMMENTS
The classical Gilbreath-Proth Conjecture is that when iterated absolute differences are formed from the sequence of primes, the leading diagonal is 2,1,1,1,1,1,1,1,1,... (see A036262). This is an analog for the lucky numbers sequence.
This is the Gilbreath transform of the lucky numbers (cf. A362451). It appears that apart from the initial term, all the other terms are 0 or 2 (compare A362460). - N. J. A. Sloane, May 07 2023
REFERENCES
Henry Gould, Gilbreath-Proth type sequence generated from Lucky numbers, unpublished.
FORMULA
a(n) = A254967(n,0). - Reinhard Zumkeller, Feb 11 2015
MATHEMATICA
nmax = 104; (* index of last term *)
imax = 400; (* max index of initial lucky array L *)
L = Table[2 i + 1, {i, 0, imax}];
For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];
T[n_, n_] := If[n + 1 <= Length[L], L[[n + 1]], Print["imax should be increased"]; 0];
T[n_, k_] := T[n, k] = Abs[T[n, k + 1] - T[n - 1, k]];
a[n_] := T[n, 0];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 22 2021 *)
A000959[upto_]:=Module[{s=2, a=Range[1, upto, 2]}, While[s<Length[a]&&a[[s]]<=Length[a], a=Drop[a, {a[[s]], -1, a[[s++]]}]]; a];
A054978[upto_]:=Module[{d=A000959[upto]}, Join[{1}, Table[First[d=Abs[Differences[d]]], Length[d]-1]]];
A054978[1000] (* Uses lucky numbers up to 1000 *) (* Paolo Xausa, May 11 2023 *)
PROG
(Haskell)
a054978 n = a054978_list !! n
a054978_list = map head $ iterate
(\lds -> map abs $ zipWith (-) (tail lds) lds) a000959_list
-- Reinhard Zumkeller, Feb 10 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Henry Gould, May 29 2000
EXTENSIONS
More terms from Naohiro Nomoto, Jun 16 2001
STATUS
approved