A335329 Primes p of the form 4k+1 such that the sum up to p of the primes of the same form is a square.

29, 61, 197, 11789, 7379689, 161409881, 14881142931617, 34041319775377

Offset

1,1

Links

Table of n, a(n) for n=1..8.

Example

5+13+17+29 = 64 = 8^2.
5+...+161409881 = 354203842652416 = 18820304^2.

Mathematica

s=0; Select[Prime@ Range[10^9], Mod[#, 4]==1 && IntegerQ@ Sqrt[s+=#] &] (* Robert Price, Sep 10 2020 *)
Module[{nn=74*10^5, k, a}, k=Select[Prime[Range[nn]], Mod[#-1, 4]==0&]; a=Accumulate[ k]; Select[ Thread[ {k, a}], IntegerQ[Sqrt[#[[2]]]]&]][[;; , 1]] (* The program generates the first five terms of the sequence. *) (* Harvey P. Dale, Jul 19 2024 *)

Prog

(UBASIC)
   10   'S1=sum of primes 4k+1, S1=sum of primes 4k+1
   20   'is S1 a square?
   30   S1=0:P=2:PM=2^32-10:K=1
   40   P=nxtprm(P):K=K+1:if P>PM then end
   50   if P@4=3 then goto 40
   60   S1=S1+P:SS1=isqrt(S1)
   70   if SS1*SS1=S1 then print K; P; S1; SS1; 1
   80   goto 40
(PARI) s=0; forprime(p=5, 10^9, if(p%4==1, s+=p; if(issquare(s), print1(p, ", ")))) \\ Hugo Pfoertner, Jun 02 2020

Crossrefs

Cf. A033998.

Sequence in context: A122114 A173032 A142047 * A046275 A092222 A338529

Adjacent sequences: A335326 A335327 A335328 * A335330 A335331 A335332

Keyword

nonn,more

Author

Carlos Rivera, Jun 01 2020

Extensions

a(7) and a(8) from Martin Ehrenstein using Kim Walisch's primesieve, Jan 09 2021

Status

approved