A064397 Numbers k such that prime(k) + prime(k+1) is a square.

7, 15, 20, 61, 152, 190, 293, 377, 492, 558, 789, 919, 942, 1768, 2343, 2429, 2693, 2952, 3136, 3720, 4837, 5421, 5722, 6870, 7347, 8126, 8193, 9465, 9857, 9927, 10410, 10483, 10653, 12685, 13763, 13955, 16033, 16342, 17859, 18367, 18474

Offset: 1

Author

Jason Earls, Sep 29 2001

Keywords

• nonn

Examples

			For k=15: prime(15) = 47 and prime(16) = 53, 47 + 53 = 100 = 10^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Crossrefs

Cf. A061275 (the primes), A062703 (squares), A074924 (square root of sum), A000720.

Cf. A076305 (3 primes), A072849 (4 primes), A166255 (70 primes), A166261 (120 primes).

Programs

Magma

[n: n in [0..50000]| IsSquare(NthPrime(n) +NthPrime(n+1))]; // Vincenzo Librandi, Apr 06 2011

Mathematica

lst={};Do[p1=Prime[n];p2=Prime[n+1];q=(p1+p2)^0.5;If[q==IntegerPart[q], AppendTo[lst, n]], {n, 1, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)

PARI

j=[]; for(n=1,30000,x=prime(n)+prime(n+1); if(issquare(x),j=concat(j,n))); j

PARI

{ n=0; default(primelimit, 8500000); for (m=1, 10^9, if (issquare(prime(m) + prime(m + 1)), write("b064397.txt", n++, " ", m); if (n==175, break)) ) } \\ Harry J. Smith, Sep 13 2009

Formula

a(n) = A000720(A061275(n)). - Amiram Eldar, Jun 28 2024

a(n) >> n^2/log^2 n. - Charles R Greathouse IV, Mar 08 2025