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 A205386.)
(See the program at A205394.)
(See the program at A205394.)
(See the program at A205402.)
(See the program at A205450.)
(See the program at A205450.)
1 divides 2^5-1^5 -> k=2, j=1 2 divides 3^5-1^5 -> k=3, j=1 3 divides 4^5-1^5 -> k=4, j=1 4 divides 4^5-2^5 -> k=4, j=2 5 divides 6^5-1^5 -> k=6, j=1 6 divides 7^5-1^5 -> k=7, j=1
N:= 100: # for a(1)..a(N)
V:= Vector(N):
count:= 0:
for k from 1 while count < N do
for j from 1 to k-1 while count < N do
Q:= select(t -> t <= N and V[t] = 0, numtheory:-divisors(k^5-j^5));
if Q <> {} then
newcount:= nops(Q);
count:= count + newcount;
V[convert(Q,list)]:= k;
fi
od od:
convert(V,list); # Robert Israel, May 14 2021
s = Table[n^4, {n, 1, 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 *)
Array[(k=1;While[FreeQ[Mod[Table[k^5-j^5,{j,k-1}],#],0],k++];k)&,100] (* Giorgos Kalogeropoulos, May 14 2021 *)
The first six terms match these differences: s(5)-s(3) = 8-3 = 5 = 5*1 s(6)-s(3) = 13-3 = 10 = 5*2 s(6)-s(5) = 13-8 = 5 = 5*1 s(7)-s(1) = 21-1 = 20 = 5*4 s(9)-s(4) = 55-5 = 50 = 5*10 s(10)-s(8) = 89-34 = 55 =5*11
s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}] (* A204922 *)
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] = Delete[w[n], Position[w[n], 0]]
c = 5; t = d[c] (* A205851 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}] (* A205852 *)
Table[j[n], {n, 1, z2}] (* A205853 *)
Table[s[k[n]]-s[j[n]], {n,1,z2}] (* A205854 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205855 *)
The first six terms match these differences: s(5)-s(2) = 8-2 = 6 = 6*1 s(6)-s(1) = 13-1 = 12 = 6*2 s(7)-s(3) = 21-3 = 18 = 6*3 s(9)-s(1) = 55-1 = 54 = 6*9 s(9)-s(6) = 55-13 = 42 = 6*7 s(10)-s(4) = 89-5 = 84 =6*14
s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}] (* A204922 *)
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] = Delete[w[n], Position[w[n], 0]]
c = 6; t = d[c] (* A205856 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}] (* A205857 *)
Table[j[n], {n, 1, z2}] (* A205858 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}] (* A205859 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}] (* A205860 *)
The first six terms match these differences: s(5)-s(1) = 8-1 = 7 = 7*1 s(8)-s(6) = 34-13 = 21 = 7*3 s(9)-s(6) = 55-13 = 42 = 7*6 s(9)-s(8) = 55-34 = 21 = 7*3 s(10)-s(4) = 89-5 = 84 = 7*12 s(13)-s(6) = 377-13 = 364 =7*52
s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}] (* A204922 *)
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] = Delete[w[n], Position[w[n], 0]]
c = 7; t = d[c] (* A205861 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}] (* A205862 *)
Table[j[n], {n, 1, z2}] (* A205863 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205864 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}] (* A205865 *)
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