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A033997 Numbers n such that sum of first n primes is a square. 10
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
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
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 March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)