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.
mkl:= proc(i,l) local ll, mn, x; ll:= applyop (x->x+1, irem (i,5)+1, l); mn:= min (ll[]); `if` (mn=0, ll, map (x->x-mn, ll)) end: g:= proc (n,i,t) if n<0 then 0 elif n=0 then `if` (t[2]=t[5] and t[5]<=t[1] and t[1]<=t[3] and t[3]=t[4],1,0) elif i=0 then 0 elif i=1 then g (0, 0, [t[1], t[2]+n, t[3], t[4], t[5]]) elif i=2 then `if` (t[3]>t[4], 0, g (n-2*(t[4]-t[3]), 1, [t[1], t[2], t[4], t[4], t[5]])) else g(n,i,t):= g (n,i-1,t) +g (n-i,i, mkl(i,t)) fi end: a:= n-> g(5*n, 5*n, [0,0,0,0,0]): seq(a(n), n=1..15); # Alois P. Heinz, Jul 02 2009
mkl[i_, l_] := Module[{ll, mn, x}, ll = MapAt[#+1&, l, Mod[i, 5]+1]; mn = Min[ll]; If[mn==0, ll, Map[#-mn&, ll]]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0, If[t[[2]] == t[[5]] && t[[5]] <= t[[1]] && t[[1]] <= t[[3]] && t[[3]] == t[[4]], 1, 0], i==0, 0, i==1, g[0, 0, {t[[1]], t[[2]]+n, t[[3]], t[[4]], t[[5]]}] , i==2, If[t[[3]]>t[[4]], 0, g[n-2*(t[[4]]-t[[3]]), 1, {t[[1]], t[[2]], t[[4]], t[[4]], t[[5]]}]], True, g[n, i-1, t] + g[n-i, i, mkl[i, t]]]; a[n_] := g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)
a(3) = 13 because prime(3) = 5, the next two primes are 7 and 11, and 5*6 + 7*6 + 11*10 = 182 = 13*14.
P:= select(isprime, [2,seq(i,i=3..10^6,2)]):
R:= convert(map(p -> (p*(p-1),p*(p+1)),P),set):
f:= proc(n) local S,T,SR,i,s;
S:= {P[n]*(P[n]-1),P[n]*(P[n]+1)};
for i from n+1 do
T:= [P[i]*(P[i]-1),P[i]*(P[i]+1)];
S:= map(s -> (s+T[1],s+T[2]),S);
SR:= S intersect R;
if SR <> {} then
s:= (sqrt(1+4*min(SR))-1)/2;
if isprime(s) then return s else return s+1 fi
fi
od
end proc:
map(f, [$1..100]);
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