A131587 Sum of the squares of the first 10^n primes.

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

Offset: 0

Author

Cino Hilliard, Aug 29 2007, Oct 25 2007

Keywords

• nonn

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]

Examples

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

Links

• Bruce Garner, Table of n, a(n) for n = 0..14
• Cino Hilliard, SumprimesGmpSq.

Programs

Mathematica

Accumulate[{Prime[1]^2}~Join~Array[Total[Prime[#]^2 & /@ Range @@ {10^# + 1, 10^(# + 1)}] &, 6, 0]] (* Michael De Vlieger, Mar 14 2021 *)

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","); ) }