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 A204994.)
(See the program at A204994.)
(See the program at A205146.)
s(m) = prime(m)*prime(m+1); isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (j))); a(n) = my(k=1, x); while (!(j=isok(k, n)), k++); s(k) - s(j); \\ Michel Marcus, Jul 23 2021
The least k such that 9 divides s(k)-s(j) for some k is k=8, for which j=2: s(8)-s(2) = 20-2 = 18, so a(9)=18.
(See the program at A205402.) s[m_]:=s[m]=Floor[(m+1)^2/2]/2 A205406[n_]:=(k=2; found=False; While[!found, Do[If[Mod[d=s[k]-s[j], n]==0, found=True; Break[]], {j, k-1}]; k++]; d) nterms=100; Table[A205406[n], {n, nterms}] (* Paolo Xausa, Dec 03 2021 *)
A002620(n) = ((n^2)>>2); A002620shiftedleft(n) = A002620(1+n); A205406(n) = { my(d); for(k=2,oo, for(j=1,k-1,if(!((d=A002620shiftedleft(k)-A002620shiftedleft(j))%n),return(d)))); }; \\ Antti Karttunen, Dec 05 2021
(See the program at A205442.)
1 divides 2^2-1^1 -> k=2, j=1 2 divides 3^3-1^1 -> k=3, j=1 3 divides 2^2-1^1 -> k=2, j=1 4 divides 4^4-2^2 -> k=4, j=2
f:= proc(n) local S,k,v;
S:= {}:
for k from 1 do
v:= k &^ k mod n;
if member(v,S) then return k fi;
S:= S union {v}
od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 23 2023
s = Table[n^n, {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 *)
1 divides (2*2)!-(2*1)! -> k=2, j=1 2 divides (2*2)!-(2*1)! -> k=2, j=1 3 divides (2*3)!-(2*2)! -> k=3, j=2 4 divides (2*3)!-(2*2)! -> k=3, j=2 5 divides (2*4)!-(2*3)! -> k=4, j=3
f:= proc(n) local S,j,x;
S:= {}:
x:= 1:
for j from 1 do
x:=x*2*j*(2*j-1) mod n;
if member(x,S) then return j fi;
S:= S union {x}
od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 18 2024
s = Table[(2n)!, {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 *)
The first six terms match these differences: s(3)-s(1) = 3-1 = 2 s(4)-s(1) = 5-1 = 4 s(4)-s(3) = 5-3 = 2 s(5)-s(2) = 8-2 = 6 s(6)-s(1) = 13-1 = 12 s(6)-s(3) = 13-3 = 10
s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; 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 = 2; t = d[c] (* A205556 *)
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}] (* A205837 *)
Table[j[n], {n, 1, z2}] (* A205838 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}](* A205839 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}](* A205840 *)
(See the program at A205402.) s[m_]:=s[m]=Floor[(m+1)^2/2]/2 A205406[n_]:=(k=2;found=False;While[!found,Do[If[Mod[d=s[k]-s[j],n]==0,found=True;Break[]],{j,k-1}];k++];d) nterms=100;Table[A205406[n]/n,{n,nterms}] (* Paolo Xausa, Dec 03 2021 *)
A002620(n) = ((n^2)>>2); A002620shiftedleft(n) = A002620(1+n); A205406(n) = { my(d); for(k=2,oo, for(j=1,k-1,if(!((d=A002620shiftedleft(k)-A002620shiftedleft(j))%n),return(d)))); }; A198293(n) = (A205406(n)/n); \\ Antti Karttunen, Dec 03 2021
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