A293858 Let n be even; m = n/2 and p a prime such that p<=m with n-p nonprime. The sequence contains the successive positive maxima of values n with L = primepi(m-1)-primepi(p+1)> 0.

16, 44, 92, 148, 368, 400, 530, 688, 992, 1052, 2228, 3562, 4952, 7102, 10262, 20684, 37052, 52394, 61456, 62828, 80144, 224648, 236476, 251806, 360524, 362534, 742856, 1655152, 1872236, 2108282, 2319728, 2707118, 8561518, 12727966, 18115354, 18245438, 21572990, 54144704

Offset

1,1

Comments

Assuming the validity of Goldbach's Conjecture, there exists an integer L and a finite decreasing sequence of prime numbers P(i); i in {1,2,...,L}, such that P(L) < ... < P(2) < P(1) < m with n-P(i) not prime and n-P(L-1) prime, for P(L-1) prime.

The point {P(L-1), n-P(L-1)} is called the "minimal Goldbach point". The connotation of the word "minimal" is that this point lies on the line y = (-x + n) and sustains the shortest perpendicular distance to the line y = x, among all points {p,q} satisfying y=(-x+n) with prime p, 2 <= p <= m, such that n-p is prime.

Let L be the length of the set {P(1),P(2),..., P(L)}.

Notice that if m is prime then L=0. Also; if n-P(1) is prime then L=0.

Links

Gilmar Rodriguez Pierluissi, Table of n, a(n) for n = 1..44

J.-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.

Gilmar Rodriguez Pierluissi, Mathematica notebook (version 11.2) with examples for sequence A293858.

Gilmar Rodriguez Pierluissi, Adobe PDF file showing content of Mathematica notebook (in case that the reader does not have the Mathematica software available).

Example

For n=16, previous prime of m is 7; (n-7) is not prime; previous prime of 7 is 5; n-5 is prime; L=Length({7})=1.
For n=44, previous prime of m is 19; (n-19)is not prime; previous prime of 19 is 17; n-17 is not prime; previous prime of 17 is 13; (n-13) is prime; L=Length({19, 17})= 2.

Mathematica

PreviousPrime[n_]:=NextPrime[n, -1]
L[n_?EvenQ]:=Module[{m=n/2}, If[PrimeQ[m], l=0, l=Length[Drop[Most@NestWhileList[PreviousPrime, m, !PrimeQ[n-#]&], 1]]]; l]
f[n_]:=For[m=n/2, True, m--, Return[L[n]]]; For[n=16; max=-1, True, n+=2, If[f[n]>max, Print[n]; max=f[n]]]

Prog

(PARI) f(n) = {len = 0; m = n/2; if (isprime(m), return (0)); p = precprime(m-1); while (1, if (isprime(n-p), return (len)); p = precprime(p-1); len ++; ); }
lista(nn) = {lmax = 0; forstep (n=2, nn, 2, newl = f(n); if (newl > lmax, print1(n, ", "); lmax = newl); ); } \\ Michel Marcus, Oct 22 2017

Crossrefs

Cf. A065978.

Sequence in context: A238255 A210375 A173560 * A258547 A211573 A211582

Adjacent sequences: A293855 A293856 A293857 * A293859 A293860 A293861

Keyword

nonn

Author

Gilmar Rodriguez Pierluissi, Oct 17 2017

Status

approved