OFFSET
1,2
COMMENTS
Conjecture: Each positive rational number can be written as (prime(k) + prime(k+1))/(prime(m) + prime(m+1)), where k and m are positive integers.
This has been verified for rational numbers a/b with 1 <= a,b <= 1000. For example,
511/587 = (1396*511)/(1396*587) = 713356/819452 with 713356 = prime(30500) + prime(30501) and 819452 = prime(34587) + prime(34588).
The above conjecture is weaker than the strong conjecture in A259540, for, if prime(m) and prime(m+1) = prime(m) + 2 are twin prime then prime(m) + prime(m+1) = 2*k, where k = prime(m) + 1 with k - 1 and k + 1 both prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310.
EXAMPLE
a(4) = 6 with 4*(prime(6) + prime(7)) = 4*(13 + 17) = 120 = 59 + 61 = prime(17) + prime(18).
MATHEMATICA
p[n_]:=p[n]=Prime[n]; S[n_]:=S[n]=p[n]+p[n+1];
f[n_]:=f[n]=Sum[If[S[k]<=n&&S[k+1]>n, k, 0], {k, 1, PrimePi[n/2]}];
YQ[n_]:=YQ[n]=S[f[n]]==n;
tab={}; Do[m=1; Label[bb]; If[YQ[n*S[m]], tab=Append[tab, m]; Goto[aa]]; m=m+1; Goto[bb]; Label[aa], {n, 1, 100}]; Print[tab]
PROG
(PARI) isok(k) = my(p=2, q=nextprime(p+1)); while(p+q <= k, if (p+q==k, return(1)); p=q; q=nextprime(q+1));
a(n) = my(m=1); while(!isok(n*(prime(m) + prime(m+1))), m++); m; \\ Michel Marcus, Oct 15 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 15 2025
STATUS
approved
