A054978 Obtained from sequence of lucky numbers (A000959) by taking iterated absolute value differences of terms and extracting the leading diagonal.

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

Author

Henry Gould, May 29 2000

Keywords

• nonn
• easy
• nice

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• Henry Gould, Gilbreath-Proth type sequence generated from Lucky numbers, unpublished.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Reinhard Zumkeller)
• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A000959, A036262, A054977.

Cf. also A254967, A362451, A362460, A362461, A362462.

Programs

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

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[sA054978[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 *)

Formula

a(n) = A254967(n,0). - Reinhard Zumkeller, Feb 11 2015

Extensions

More terms from Naohiro Nomoto, Jun 16 2001