OFFSET
1,1
COMMENTS
Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q and u/v != x, and |x - p/q| <= |x - p'/q| for all primes p' != x*q. For some choices of (rational) x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). [Corrected and edited by M. F. Hasler, May 18 2026; corrected by Pontus von Brömssen, Jun 08 2026]
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x Lower POBA Upper POBA POBA
Note that the definitions in the first remark explicitly exclude p = x*q, which would give |x - p/q| = 0. That is, if for given q (and rational x), p = x*q is prime, the "approximation" p/q must be ignored, and only larger and/or smaller primes p may be considered, see EXAMPLE. - M. F. Hasler, May 19 2026
FORMULA
a(2n) = A006512(n+1), a(2n+1) = a(2n) - 2 = A001359(n+1), for all n >= 2; i.e., after the three initial terms, the sequence lists twin primes > 7 with the larger of each pair coming first. - M. F. Hasler, May 19 2026
EXAMPLE
The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.
Indeed, for x = 1, we cannot use p = q = 2 (which would yield x - p/q = 0 and end the sequence of approximations), but have to use the next best prime p = 3; thereafter, for q = 3, we may not use p = 3 but have to use p = 2 instead. Then, for q = 5, we have the equally good p = 7 (7/5 = 1.4) and p = 3 (3/5 = 0.6), but both are worse than the previous approximation 2/3 = 0.666..., so we skip them and turn to q = 7 where p = 5 gives the better p/q = 0.7142857... We see that from that point on, the sequence of POBAs is p/q, q'/p', p'/q', q"/p", p"/q", ... with twin prime pairs (p, q = p+2). - M. F. Hasler, May 19 2026
MATHEMATICA
x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
Numerator[tL] (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU] (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y] (* A265759 *)
Denominator[y] (* A265760 *)
PROG
(PARI) A265759_first(N)=apply(numerator, POBA(1, N))
POBA(x=1, N=20, best=9, L=List())={ forprime(q=2, , my(p=x*q, pL=precprime(p-!frac(p)), dL=p-pL, pR=nextprime(p+!frac(p)), dR=pR-p); min(dR, dL) < best*q || next; dL==dR && listput(L, pR/q); listput(L, if(dR < dL, pR, pL)/q); #L < N || break; best=abs(L[#L]-x)); Vec(L)} \\ M. F. Hasler, May 18 2026
(Python) A265759_first = lambda N: [q for (p, q), i in zip(POBA(1), range(N))] # see A265812 for POBA(). - M. F. Hasler, May 18 2026
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Dec 15 2015
STATUS
approved