A347925 - OEIS

%I #17 Oct 25 2021 14:33:32

%S 1,1,1,1,1,3,6,30,180,1260,181440,1814400,19958400,239500800,

%T 3113510400,43589145600,43589145600,653837184000,177843714048000,

%U 177843714048000,1600593426432000,1216451004088320000,25545471085854720000,25545471085854720000

%N a(n) is the lowest common denominator of n-th Gilbreath polynomial.

%C Let S=(p_1, ..., p_n) be the ordered sequence of the first n prime numbers. The n-th Gilbreath polynomial is defined as the polynomial P_n such that the x-th term of the upper bound Gilbreath sequence of S, U(S)_x, is U(S)_x=2^(n+x-1)+P_n where P_n = Sum_{i=1..n} T(n,i)*x^(i-1)/a(n).

%H Riccardo Gatti, <a href="https://www.preprints.org/manuscript/202003.0145">Gilbreath Sequences and Proof of Conditions for Gilbreath Conjecture</a>, Preprints 2020, 2020030145.

%H Riccardo Gatti, <a href="https://github.com/gttrcr/ResearchCode/blob/main/OEIS/A347925.cs">Program for the generation of the m-th Gilbreath polynomial calling GenMthGilbreathPolynomial(m)</a>

%H A. M. Odlyzko, <a href="https://doi.org/10.1090/S0025-5718-1993-1182247-7">Iterated absolute values of differences of consecutive primes</a>, Math. Comp. 61 (1993), 373-380.

%e The lowest common denominator of P_6 is a(6)=3, in fact P_6 = (-57 - 55x - 15x^2 - 2x^3)/3. The x-th term of the upper bound Gilbreath sequence of S=(p_1, ..., p_6) = (2, 3, 5, 7, 11, 13) is U(S)_x = 2^(x+5) + (-57 - 55x - 15x^2 - 2x^3)/3.

%o (PARI) polynomialfit(data) = Pol(Vecrev(matsolve(matrix(#data,#data,i,j,i^(j-1)), data~))); \\ from _David A. Corneth_

%o isg(v, k) = {my(w = concat(v, k), vd = w); for (i=1, #w-1, vd = vector(#vd-1, k, abs(vd[k+1] - vd[k])); if (vd[1] != 1, return (0));); return (1);}

%o nextx(v) = {my(k = nextprime(nextprime(vecmax(v)+1)+1)); while (isg(v, k), k+=2); k-=2;}

%o a(n) = {my(vp = primes(n), v = List()); for (i=1, n, my(x = nextx(vp)); vp = concat(vp, x); listput(v, x);); v = Vec(v); my(cp = Vecrev(polynomialfit(vector(#v, k, v[k] - 2^(k+n-1))))); lcm(apply(denominator, cp)); } \\ _Michel Marcus_, Sep 20 2021

%Y Cf. A347924.

%K nonn

%O 1,6

%A _Riccardo Gatti_, Sep 20 2021