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A383504 Sum of next a(n) successive prime squares is prime.

%I A383504 #75 Jun 09 2025 14:42:07
%S A383504 2,5,7,25,11,13,7,7,35,7,19,31,25,11,5,19,5,11,5,139,37,17,19,19,13,5,
%T A383504 7,7,19,13,11,5,7,11,5,5,5,13,47,43,5,23,13,11,11,79,31,35,5,5,25,5,
%U A383504 37,95,31,13,43,17,5,35,17,23,11,41,59,7,47,5,13,7,11
%N A383504 Sum of next a(n) successive prime squares is prime.
%C A383504 Group the primes such that the sum of squares of members of each group is a prime, and each successive group is as short as possible.
%C A383504 Apart from a(1)=2, a(n) is odd and not a multiple of 3.
%H A383504 Abhiram R Devesh, <a href="/A383504/b383504.txt">Table of n, a(n) for n = 1..10000</a>
%e A383504 Primes, their squares, and the lengths of blocks which sum to a prime begin,
%e A383504   primes  2, 3,  5,  7,  11,  13,  17,  19, ...
%e A383504   squared 4, 9, 25, 49, 121, 169, 289, 361, ...
%e A383504           \--/  \-------------------/  \--- ...
%e A383504   sum      13            653
%e A383504   a(n) =    2             5
%p A383504 i:= 0: p:= 0: t:= 0: count:= 0: R:= NULL:
%p A383504 while count < 100 do
%p A383504   p:= nextprime(p);
%p A383504   i:= i+1;
%p A383504   t:= t + p^2;
%p A383504   if isprime(t) then
%p A383504     R:= R, i; count:= count+1; i:= 0; t:= 0;
%p A383504   fi
%p A383504 od:
%p A383504 R; # _Robert Israel_, May 25 2025
%t A383504 p=1;s={};Do[c=0;sm=0;While[!PrimeQ[sm],sm=sm+Prime[p]^2;p++;c++];AppendTo[s,c],{n,71}];s (* _James C. McMahon_, Jun 09 2025 *)
%o A383504 (Python)
%o A383504 from itertools import count, islice
%o A383504 from sympy import isprime, nextprime
%o A383504 def agen(): # generator of terms
%o A383504     s, i, p = 0, 1, 2
%o A383504     while True:
%o A383504         while not(isprime(s:=s+p**2)): i, p = i+1, nextprime(p)
%o A383504         yield i
%o A383504         s, i, p = 0, 1, nextprime(p)
%o A383504 print(list(islice(agen(), 71))) # _Michael S. Branicky_, May 23 2025
%Y A383504 Cf. A001248, A073684 (sum of successive primes), A384161 (sum of successive prime cubes).
%K A383504 nonn
%O A383504 1,1
%A A383504 _Abhiram R Devesh_, May 18 2025