OFFSET
1,1
COMMENTS
It appears that the only square in this sequence is 4. Checked 10^11 terms. a(10^11) = 247754953701579144582110673365391267. - T. D. Noe, Sep 06 2005
a(2n-1) is divisible by 2, a(3n+1) is divisible by 3, a(4n-3) is divisible by 4, a(6n+1) is divisible by 6, a(8n-3) is divisible by 8, a(12n+1) is divisible by 12, a(24n-11) is divisible by 24. - Alexander Adamchuk, Jun 15 2006
The sequence is best looked at in base 12, with X for 10 and E for 11: 4, 11, 32, 73, 154, 275, 476, 717, X98, 1479, 1E3X, 289E, 3860, 4941, 6082, 7823, 9844, EX25, 12546, 15447, 18548, 20089, 2406X, 2876E, 320E0, 37E91, 42152, 488E3, 53754, 5E015, 68416, 76337, 85178, 94399, X51EX, E643E, 108760, 120001. Since the squares of all primes greater than 3 are always 1 mod 12, the sequence obeys the rule a(n) mod 12 = (n-1) mod 12 for n>=2. The rule gives a(2n-1) = (2n-2) mod 12 and so a(2n-1) must be divisible by 2. a(3n+1) = (3n) mod 12 so a(3n+1) is divisible by 3. The other rules are proved similarly. Remember: base 12 is a research tool! - Walter Kehowski, Jun 24 2006
For all primes p > 3, we have p^2 == 1 (mod m) for m dividing 24 (and only these m). Using a covering argument, it is not hard to show that all terms except a(24k+13) are nonsquares. Hence in the search for square a(n), only 1 out of every 24 terms needs to be checked. - T. D. Noe, Jan 23 2008
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..5000
Carlos Rivera, Puzzle 128: Sum of consecutive squared primes a square, The Prime Puzzles & Problems Connection.
Vladimir Shevelev, Asymptotics of sum of the first n primes with a remainder term
Eric Weisstein's World of Mathematics, Prime Sums
Eric Weisstein's World of Mathematics, Prime Zeta Function
FORMULA
a(n) = Sum_{i=1..n} prime(i)^2. - Walter Kehowski, Jun 24 2006
a(n) = (1/3)*n^3*log(n)^2 + O(n^3*log(n)*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). - Vladimir Shevelev, Aug 02 2013
a(n) = a(n-1) + prime(n)^2, with a(1) = 4. - G. C. Greubel, Jan 30 2025
MAPLE
MATHEMATICA
Table[ Sum[ Prime[k]^2, {k, 1, n} ], {n, 40} ]
Accumulate[Prime[Range[40]]^2] (* Harvey P. Dale, Apr 16 2013 *)
PROG
(PARI) s=0; forprime(p=2, 1e3, print1(s+=p^2", ")) \\ Charles R Greathouse IV, Jul 15 2011
(PARI) a(n) = norml2(primes(n)); \\ Michel Marcus, Nov 26 2020
(Haskell)
a024450 n = a024450_list !! (n-1)
a024450_list = scanl1 (+) a001248_list
-- Reinhard Zumkeller, Jun 08 2015
(Magma) [&+[NthPrime(k)^2: k in [1..n]]: n in [1..40]]; // Vincenzo Librandi, Oct 11 2018
(Magma) [n le 1 select 4 else Self(n-1) + NthPrime(n)^2: n in [1..80]]; // G. C. Greubel, Jan 30 2025
(Python)
from sympy import prime, primerange
def a(n): return sum(p*p for p in primerange(1, prime(n)+1))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Apr 13 2021
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
STATUS
approved