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.
print1(1,", ");v=[1]; n=1; while(n<100, if(issquarefree(n+v[#v])&&!vecsearch(vecsort(v), n), v=concat(v, n);if(n==#v,print1(n,", ")); n=0); n++) \\ Derek Orr, Jun 09 2015
v=[1]; n=1; while(n<100, if(!issquarefree(v[#v]+n)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 08 2015
a={1}; While[Length[a]<1000, e=Last@a; s = Select[ Complement[ Range@e, a], SquareFreeQ[e^2 + #^2]&, 1]; If[s!={}, s=s[[1]], s=e+1; While[ MemberQ[a, s] || !SquareFreeQ[ e^2 + s^2], s++]]; AppendTo[a, s]]; a (* first 1000 terms, Giovanni Resta, Jun 02 2015 *)
v=[1];n=1;while(n<100,if(issquarefree(n^2+v[#v]^2)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v
a(8) = 9 because 1, 2, 3, 4, 5, 6 and 7 have already been used in the sequence, 7 + 8 = 15 is not prime power while 7 + 9 = 16 is a prime power.
N:= 100: # to get all terms before the first term > N
S:= [$2..N]:
a[1]:= 1: found:= true:
for n from 2 while found do
found:= false;
for j from 1 to nops(S) do
if ispp(a[n-1]+S[j]) then
found:= true;
a[n]:= S[j];
S:= subsop(j=NULL,S);
break
fi
od;
od:
seq(a[i],i=1..n-2); # Robert Israel, Apr 16 2017
f[s_List] := Block[{k = 1, a = s[[-1]]}, While[MemberQ[s, k] || ! PrimePowerQ[a + k], k++]; Append[s, k]]; Nest[f, {1}, 80]
Res:= 1: count:= 1: v:= 1:
Cands:= [$2..1000]:
for n from 2 do
found:= false;
for j from 1 to nops(Cands) do
if numtheory:-issqrfree(v + Cands[j]^2) then
found:= true;
if n = Cands[j] then Res:= Res, n; count:= count+1 fi;
v:= Cands[j]^2;
Cands:= subsop(j=NULL, Cands);
break
fi
od;
if not found then break fi;
od:
Res; # Robert Israel, Jul 16 2019
print1(1,", ");v=[1]; n=1; while(#v<10^3, if(issquarefree(n^2+v[#v]^2)&&!vecsearch(vecsort(v), n), if(n==#v, print1(n, ", ")); n=0); n++)
a(5) = 12 because 1, 3, 4 and 5 have already been used in the sequence, 4 + 2 = 6, 4 + 6 = 10, 4 + 7 = 11, 4 + 8 = 12, 4 + 9 = 13, 4 + 10 = 14 and 4 + 11 = 15 are not proper prime powers while 4 + 12 = 16 is a proper prime power.
N:= 2000: # to get all terms before the first where a(n)+a(n-1)>N
PP:= {seq(seq(p^j, j =2..floor(log[p](N))),p = select(isprime,[2,seq(i,i=3..floor(sqrt(N)),2)]))}:
PP:= sort(convert(PP,list)):
V:= Vector(N,datatype=integer[1],1):
A[1]:= 1; V[1]:= 0;
for n from 2 do
for pp in PP do
t:= pp - A[n-1];
if t > 0 and V[t] = 1 then
A[n]:= t; V[t]:= 0; break
fi;
od;
if not assigned(A[n]) then break fi
od:
seq(A[i],i=1..n-1); # Robert Israel, Apr 24 2017
f[s_List] := Block[{k = 1, a = s[[-1]]}, While[MemberQ[s, k] || ! (PrimePowerQ[a + k] && PrimeOmega[a + k] > 1), k++];Append[s, k]]; Nest[f, {1}, 80]
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