A033997 Numbers n such that sum of first n primes is a square.

9, 2474, 6694, 7785, 709838, 126789311423

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

Table of n, a(n) for n=1..6.

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

Carlos Rivera, Puzzle 9. Sum of first k primes is perfect square, The Prime Puzzles and Problems Connection.

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).

Cf. also A175133, A364696, A366270.

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