This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For q=3 we have (r-q)/(q-p)=2/1. Therefore, a(1)=3. For q=5: (r-q)/(q-p) = 1/1; for q = 7: (r-q)/(q-p) = 2/1; for q = 11: (r-q)/(q-p) = 1/2. Therefore, a(2)=11.
f[n_] := Block[{p = 2, q = 3, r = 5}, While[ Denominator[(r - q)/(q - p)] != n, p = q; q = r; r = NextPrime@ r]; q]; Array[f, 50]
p = 2; q = 3; r = 5; t[] = 0; While[q < 100000000, If[ t[ Denominator[(r - q)/(q - p)]] == 0, t[ Denominator[(r - q)/(q - p)]] = q]; p = q; q = r; r = NextPrime@ r]; t@# & /@ Range@100 (* _Robert G. Wilson v, Dec 11 2016 *)
a(n)=my(p=2,q=3);forprime(r=5,default(primelimit),if(denominator((r-q)/(q-p))==n,return(q));p=q;q=r)
f[n_] := Block[{p = 2, q = 3, r = 5}, While[ Numerator[(r - q)/(q - p)] != n, p = q; q = r; r = NextPrime@ r]; q]; Array[f, 36]
p = 2; q = 3; r = 5; t[] = 0; While[q < 100000000, If[ t[ Denominator[(q - p)/(r - q)]] == 0, t[ Denominator[(q - p)/(r - q)]] = q]; p = q; q = r; r = NextPrime@ r]; t@# & /@ Range@100 (* _Robert G. Wilson v, Dec 11 2016 *)
f[n_] := f[n] = Block[{p = 3, q = 5, r = 7}, While[p + n*r != (n + 1) q, p = q; q = r; r = NextPrime@ r]; q]; Array[f, 33]
p = 2; q = 3; r = 5; t[] = 0; While[p < 10^9, If[ Mod[q - p, r - q] == 0 && t[(q - p)/(r - q)] == 0, t[(q - p)/(r - q)] = q; Print[{(q - p)/(r - q), q}]]; p = q; q = r; r = NextPrime@ r]; t@# & /@ Range@45 (* _Robert G. Wilson v, Dec 10 2016 *)
a(1) = 3 because this is the first prime for which it is possible to determine the ratio between the distance to the next prime (5) and from the previous prime (2). This first ratio is 2. a(2) = 5 because the ratio between the distance to the next prime (7) and from the previous prime (3) is 1 and this ratio has not appeared before. The third element a(3) is not 7 because (11-7)/(7-5) = 2, a ratio that appeared before with a(1), so a(3) = 11 because (13-11)/(11-7) = 1/2, a ratio that did not appear before.
nmax = 720;
a = Prime[Range[nmax]];
gaps = Rest[a] - Most[a];
gapsratio = Rest[gaps]/Most[gaps];
newpindex = {}; newgratios = {}; i = 1;
While[i < Length[gapsratio] + 1,
If[Cases[newgratios, gapsratio[[i]]] == {},
AppendTo[newpindex, i + 1];
AppendTo[newgratios, gapsratio[[i]]] ];
i++];
Prime[newpindex]
p = 2; q = 3; r = 5; rtlst = qlst = {}; While[q < 10000, rt = (r - q)/(q - p); If[ !MemberQ[rtlst, rt], AppendTo[rtlst, rt]; AppendTo[qlst, q]]; p = q; q = r; r = NextPrime@ r]; qlst (* Robert G. Wilson v, Nov 30 2016 *)
For n = 1..3, A006843(n) = 1, and p,q,r have to obey the condition r-q | q-p. Thus a(1) = 5, a(2) = 11, a(3) = 17.
with(numtheory):
a:= proc(n) option remember; local k, p, q, r, pn;
pn:= ithprime(n);
for k from `if`(n=1, 1, pi(a(n-1))) do
p:= ithprime(k);
q:= ithprime(k+1);
r:= ithprime(k+2);
if denom((q-p)/(r-q)) = pn then break fi
od; q
end:
seq(a(n), n=1..10); # Alois P. Heinz, Jan 06 2011
a[n_] := a[n] = Module[{k, p, q, r, pn},
pn = Prime[n];
For[k = If[n == 1, 1, PrimePi[a[n - 1]]], True, k++,
p = Prime[k];
q = Prime[k + 1];
r = Prime[k + 2];
If [Denominator[(q - p)/(r - q)] == pn, Break[]]]; q];
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)
First 10 cases of {n,p,q,r}: {1,3,5,7}, {2,2,3,5}, {3,29,31,37}, {4,8117,8123,8147}, {5,137,139,149}, {6,197,199,211}, {7,45433,45439,45481}, {8,1931,1933,1949}, {9,521,523,541}, {10,156151,156157,156217}.
A001223 := proc(n) ithprime(n+1)-ithprime(n) ; end proc: A178942 := proc(n) option remember; local p,q,r ; if n = 1 then 3; else for q from procname(n-1)+1 do if isprime(q) then p := prevprime(q) ; r := nextprime(q) ; denom((q-p)/(r-q)) ; if % = A001223(n)/2 then return q; end if; end if; end do: end if; end proc: # R. J. Mathar, Jan 07 2011
A001223[n_] := Prime[n + 1] - Prime[n]; a[n_] := a[n] = Module[{p, q, r, d}, If[n == 1, 3, For[q = a[n - 1] + 1, True, q++, If [PrimeQ[q], p = NextPrime[q, -1]; r = NextPrime[q]; d = Denominator[(q - p)/(r - q)]; If[d == A001223[n]/2, Return[q]]]]]]; Array[a, 48] (* Jean-François Alcover, May 21 2020, after Maple *)
Row 1: 1/1 5 Row 2: 1/2 3 Row 3: 1/3 2/3 31 23 Row 4: 1/4 3/4 8123 89 Row 5: 1/5 2/5 3/5 4/5 139 7963 337 409 Row 6: 1/6 5/6 199 797 Row 7: 1/7 .. 6/7 45439 113 953 88547 293 2633 Row 8: 1/8 3/8 5/8 7/8 1933 3643 137029 13381 etc.
f[n_] := Block[{p = 2, q = 3, r = 5}, While[q != n(r - q) + p, p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff
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