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 A033997 Numbers n such that sum of first n primes is a square. 5
 9, 2474, 6694, 7785, 709838, 126789311423 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Szabolcs Tengely asks if this sequence is infinite (see Lorentz Center paper). Luca shows that this sequence is of asymptotic density 0. Cilleruelo & Luca give a lower bound. - Charles R Greathouse IV, Feb 01 2013 REFERENCES Florian Luca, On the sum of the first n primes being a square, Lithuanian Mathematical Journal 47:3 (2007), pp 243-247. LINKS Jan-Hendrik Evertse, Some open problems about Diophantine equations, Solvability of Diophantine Equations conference, Lorentz Center of Leiden University, The Netherlands. Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp. G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9 C. Rivera, PrimePuzzles.Net, Problem 9: Sum of first k primes is perfect square FORMULA a(n) = pi(A033998(n)). EXAMPLE Sum of first 9 primes is 2+3+5+7+11+13+17+19+23 = 100, which is square, so 9 is in the sequence. MATHEMATICA p = 2; s = 0; lst = {}; While[p < 10^7, s = s + p; If[ IntegerQ@ Sqrt@ s, AppendTo[lst, PrimePi@ p]; Print@ lst]; p = NextPrime@ p] (* Zak Seidov, Apr 11 2011 *) PROG (PARI) n=0; s=0; forprime(p=2, 1e6, n++; if(issquare(s+=p), print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013 CROSSREFS Cf. A000040, A033998, A061888, A061890 (associated squares). Sequence in context: A323517 A278914 A069704 * A068729 A321282 A159775 Adjacent sequences:  A033994 A033995 A033996 * A033998 A033999 A034000 KEYWORD nonn AUTHOR Calculated by Jud McCranie EXTENSIONS 126789311423 from Giovanni Resta, May 27 2003 Edited by Ray Chandler, Mar 20 2007 STATUS approved

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Last modified August 6 18:25 EDT 2020. Contains 336256 sequences. (Running on oeis4.)