A206606 Primes that can be written as a sum of a positive square and a positive cube in more than two ways.

2089, 4481, 7057, 15193, 15641, 16649, 23417, 34721, 65537, 68489, 69697, 72577, 93241, 118673, 123209, 146161, 173897, 176401, 191969, 199873, 205721, 216233, 239633, 259121, 264169, 271169, 280009, 286289, 296353, 301409, 318313, 342233, 347993, 357569, 381529, 447569, 466273, 477577, 526249, 534577

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Feb 10 2012

Keywords

• nonn

Comments

A subset of these, {65537, 93241, 191969, ..} allows this representation in more than 3 ways.

Examples

			2089 = 19^2+12^3 = 33^2+10^3 = 45^2+4^3
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3930

Crossrefs

Cf. A054402, A123364, A162930

Programs

Maple

N:= 10^6: # to get all terms <= N
for x from 1 to floor(N^(1/2)) do
  for y from 1 to floor((N-x^2)^(1/3)) do
    p:= x^2 + y^3;
    if isprime(p) then
      if assigned(R[p]) then R[p]:= R[p]+1
      else R[p]:= 1
      fi
    fi
  od
od:
sort(map(op,select(t -> R[op(t)]>2, [indices(R)]))); # Robert Israel, Mar 21 2017

Mathematica

t={}; Do[Do[AppendTo[t,n^2+m^3],{n,300}],{m,300}]; t=Sort[t]; t3={}; Do[If[t[[n]]==t[[n+2]]&&PrimeQ[t[[n]]],AppendTo[t3,t[[n]]]],{n,Length[t]-2}]; t3; f1[l_]:=Module[{t={}},Do[If[l[[n]]!=l[[n+1]],AppendTo[t,l[[n]]]],{n,Length[l]-1}];t]; (*ExtractSingleTermsOnly*) f1[t3] (* or *)
mx = 10^6; First /@ Sort@ Select[ Tally[ Join @@ Reap[(Sow@ Select[#^3 + Range[ Sqrt[mx - #^3]]^2, PrimeQ]) & /@ Range[mx^(1/3)]][[2, 1]]], #[[2]]>2 &] (* faster, Giovanni Resta, Mar 21 2017 *)

Extensions

More terms from Robert Israel, Mar 21 2017