

A131587


Sum of the squares of the first 10^n primes.


0



4, 2397, 8384727, 19053119163, 34099597499091, 53251529659694763, 76304519151822049179, 103158861357874372432083, 133759354162117403400944283, 168072405102068540986037048787, 206076219788796447007218742841043, 247754953701579144582110673365391267
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OFFSET

0,1


COMMENTS

Except for the first term, these numbers are divisible by 3. This follows from the fact that all primes are of the form 3m+1 or 3m+2 and the square of either of these forms is of the form 3h+1.
Then for the first 10 primes, the sum of the squares becomes 4+9+3h*8+8 = 21+3h*8, which is divisible by 3.
By induction, assuming that the sum of the squares of the first 10^n primes is divisible by 3, then the difference between the sum for n+1 and the sum for n is (3*h+1)*(10^(n+1)  10^n) = (3*h+1)*(9*10^n), which is divisible by 3. [Comments corrected by Paul R. F. Schumacher, Mar 16 2008]


LINKS

Table of n, a(n) for n=0..11.
C. Hilliard, SumprimesGmpSq.


EXAMPLE

The sum of squares of the first 10^1 primes = 2397, the second entry in the sequence.


PROG

(PARI) sumprimesq(n, b) = { local(x, y, s, a); for(y=0, n, s=0; for(x=1, b^y, s+=prime(x)^2; ); print1(s", "); ) }


CROSSREFS

Sequence in context: A047676 A280790 A079187 * A066850 A066837 A275683
Adjacent sequences: A131584 A131585 A131586 * A131588 A131589 A131590


KEYWORD

nonn


AUTHOR

Cino Hilliard, Aug 29 2007, Oct 25 2007


STATUS

approved



