cp's OEIS Frontend

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.

%I A293858 #88 Nov 27 2021 11:07:09
%S A293858 16,44,92,148,368,400,530,688,992,1052,2228,3562,4952,7102,10262,
%T A293858 20684,37052,52394,61456,62828,80144,224648,236476,251806,360524,
%U A293858 362534,742856,1655152,1872236,2108282,2319728,2707118,8561518,12727966,18115354,18245438,21572990,54144704
%N 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.
%C A293858 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.
%C A293858 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.
%C A293858 Let L be the length of the set {P(1),P(2),..., P(L)}.
%C A293858 Notice that if m is prime then L=0.  Also; if n-P(1) is prime then L=0.
%H A293858 Gilmar Rodriguez Pierluissi, <a href="/A293858/b293858.txt">Table of n, a(n) for n = 1..44</a>
%H A293858 J.-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, <a href="https://doi.org/10.1090/S0025-5718-1993-1202609-9">An upper bound in Goldbach's problem</a>, Math. Comp. 61 (1993), 209-213.
%H A293858 Gilmar Rodriguez Pierluissi, <a href="/A293858/a293858_2.nb">Mathematica notebook (version 11.2) with examples for sequence A293858.</a>
%H A293858 Gilmar Rodriguez Pierluissi, <a href="/A293858/a293858_2.pdf">Adobe PDF file showing content of Mathematica notebook (in case that the reader does not have the Mathematica software available).</a>
%e A293858 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.
%e A293858 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.
%t A293858 PreviousPrime[n_]:=NextPrime[n, -1]
%t A293858 L[n_?EvenQ]:=Module[{m=n/2},If[PrimeQ[m],l=0,l=Length[Drop[Most@NestWhileList[PreviousPrime,m,!PrimeQ[n-#]&],1]]];l]
%t A293858 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]]]
%o A293858 (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 ++;);}
%o A293858 lista(nn) = {lmax = 0; forstep (n=2, nn, 2, newl = f(n); if (newl > lmax, print1(n, ", "); lmax = newl););} \\ _Michel Marcus_, Oct 22 2017
%Y A293858 Cf. A065978.
%K A293858 nonn
%O A293858 1,1
%A A293858 _Gilmar Rodriguez Pierluissi_, Oct 17 2017