|
%I #33 Sep 25 2023 19:19:51
%S 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,
%T 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,
%U 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
%N Obtained from sequence of lucky numbers (A000959) by taking iterated absolute value differences of terms and extracting the leading diagonal.
%C 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.
%C 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
%D Henry Gould, Gilbreath-Proth type sequence generated from Lucky numbers, unpublished.
%H Paolo Xausa, <a href="/A054978/b054978.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Reinhard Zumkeller)
%H <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%F a(n) = A254967(n,0). - _Reinhard Zumkeller_, Feb 11 2015
%t nmax = 104; (* index of last term *)
%t imax = 400; (* max index of initial lucky array L *)
%t L = Table[2 i + 1, {i, 0, imax}];
%t For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];
%t T[n_, n_] := If[n + 1 <= Length[L], L[[n + 1]], Print["imax should be increased"]; 0];
%t T[n_, k_] := T[n, k] = Abs[T[n, k + 1] - T[n - 1, k]];
%t a[n_] := T[n, 0];
%t Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Sep 22 2021 *)
%t 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];
%t A054978[upto_]:=Module[{d=A000959[upto]},Join[{1},Table[First[d=Abs[Differences[d]]],Length[d]-1]]];
%t A054978[1000] (* Uses lucky numbers up to 1000 *) (* _Paolo Xausa_, May 11 2023 *)
%o (Haskell)
%o a054978 n = a054978_list !! n
%o a054978_list = map head $ iterate
%o (\lds -> map abs $ zipWith (-) (tail lds) lds) a000959_list
%o -- _Reinhard Zumkeller_, Feb 10 2015
%Y Cf. A000959, A036262, A054977.
%Y Cf. also A254967, A362451, A362460, A362461, A362462.
%K nonn,easy,nice
%O 0,2
%A _Henry Gould_, May 29 2000
%E More terms from _Naohiro Nomoto_, Jun 16 2001
|
