A347925 a(n) is the lowest common denominator of n-th Gilbreath polynomial.

1, 1, 1, 1, 1, 3, 6, 30, 180, 1260, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 43589145600, 653837184000, 177843714048000, 177843714048000, 1600593426432000, 1216451004088320000, 25545471085854720000, 25545471085854720000

Offset

1,6

Comments

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).

Links

Table of n, a(n) for n=1..24.

Riccardo Gatti, Gilbreath Sequences and Proof of Conditions for Gilbreath Conjecture, Preprints 2020, 2020030145.

Riccardo Gatti, Program for the generation of the m-th Gilbreath polynomial calling GenMthGilbreathPolynomial(m)

A. M. Odlyzko, Iterated absolute values of differences of consecutive primes, Math. Comp. 61 (1993), 373-380.

Example

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.

Prog

(PARI) polynomialfit(data) = Pol(Vecrev(matsolve(matrix(#data, #data, i, j, i^(j-1)), data~))); \\ from David A. Corneth
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); }
nextx(v) = {my(k = nextprime(nextprime(vecmax(v)+1)+1)); while (isg(v, k), k+=2); k-=2; }
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

Crossrefs

Cf. A347924.

Sequence in context: A157534 A372024 A133799 * A262022 A088436 A088506

Adjacent sequences: A347922 A347923 A347924 * A347926 A347927 A347928

Keyword

nonn

Author

Riccardo Gatti, Sep 20 2021

Status

approved