cp's OEIS Frontend

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

%I A131587 #18 Mar 14 2021 11:44:10
%S A131587 4,2397,8384727,19053119163,34099597499091,53251529659694763,
%T A131587 76304519151822049179,103158861357874372432083,
%U A131587 133759354162117403400944283,168072405102068540986037048787,206076219788796447007218742841043,247754953701579144582110673365391267
%N A131587 Sum of the squares of the first 10^n primes.
%C A131587 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.
%C A131587 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.
%C A131587 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]
%H A131587 Bruce Garner, <a href="/A131587/b131587.txt">Table of n, a(n) for n = 0..14</a>
%H A131587 Cino Hilliard, <a href="http://docs.google.com/Doc?docid=dgpq9w4b_14d29xkr&amp;hl=en">SumprimesGmpSq</a>.
%e A131587 The sum of squares of the first 10^1 primes = 2397, the second entry in the sequence.
%t A131587 Accumulate[{Prime[1]^2}~Join~Array[Total[Prime[#]^2 & /@ Range @@ {10^# + 1, 10^(# + 1)}] &, 6, 0]] (* _Michael De Vlieger_, Mar 14 2021 *)
%o A131587 (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","); ) }
%K A131587 nonn
%O A131587 0,1
%A A131587 _Cino Hilliard_, Aug 29 2007, Oct 25 2007