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.
(See the program at A204932.)
Let s(k)=prime(k). As in A204890, the ordering of differences s(k)-s(j), follows from the arrangement shown here: k...........1..2..3..4..5...6...7...8...9 s(k)........2..3..5..7..11..13..17..19..23 ... s(k)-s(1)......1..3..5..9..11..15..17..21..27 s(k)-s(2).........2..4..8..10..14..16..20..26 s(k)-s(3)............2..6..8...12..14..18..24 s(k)-s(4)...............4..6...10..12..16..22 ... least (k,j) such that 1 divides s(k)-s(j) for some j is (2,1), so a(1)=2. least (k,j) such that 2 divides s(k)-s(j): (3,2), so a(2)=3. least (k,j) such that 3 divides s(k)-s(j): (3,1), so a(3)=3.
s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 50;
Table[s[n], {n, 1, 30}] (* A000040 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j],
{k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}] (* A204890 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n],
Position[w[n], 0]]]
Table[d[n], {n, 1, z2}] (* A204891 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}] (* A204892 *)
Table[j[n], {n, 1, z2}] (* A204893 *)
Table[s[k[n]], {n, 1, z2}] (* A204894 *)
Table[s[j[n]], {n, 1, z2}] (* A204895 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A204896 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204897 *)
(* Program 2: generates A204892 and A204893 rapidly *)
s = Array[Prime[#] &, 120];
lk = Table[NestWhile[# + 1 &, 1, Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1, Length[s]}]
Table[NestWhile[# + 1 &, 1, Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &], {j, 1, Length[lk]}]
(* Peter J. C. Moses, Jan 27 2012 *)
a(n)=forprime(p=n+2,,forstep(k=p%n,p-1,n,if(isprime(k), return(primepi(p))))) \\ Charles R Greathouse IV, Mar 20 2013
a(1)=s(2)-s(1)=2!-1!=1 a(2)=s(3)-s(1)=3!-1!=5 a(3)=s(3)-s(2)=3!-2!=4 a(4)=s(4)-s(1)=4!-1!=23
(See the program at A204932.)
f:= proc(n) local t,k,V;
t:= 1:
for k from 1 do
t:= t*k mod n;
if assigned(V[t]) then return V[t]! else V[t]:= k fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 10 2024
(See the program at A204932.)
(See the program at A204932.)
(See the program at A204932.)
(See the example at A204932.)
(See the program at A204932.)
(See the program at A204979.)
(See the program at A204932.)
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